#Probabilités et Simulation - Devoir n°1 Lombard Myriam RICM4
##Exercice 1 : Question préliminaire à propos d’estimation
###Q1.1
Je suppose que \(max_i\) est la fonction maximum, donc que M est la valeur maximale tirée parmi l’ensemble de \(X_i\). Cet estimateur permet de se rapprocher le plus de a parmi tous les \(X_i\). Avec les paramètres (a, n) = (16, 5), ma valeur pour a est petite, mais ma valeur pour n l’est plus encore (ce qui est normal), je pense donc que cet estimateur ne va pas permettre de trouver la valeur de a à chaque coup. Je pense qu’au plus n est supérieur à a, au plus la probabilité que la valeur M soit égale à a est élevée.
set.seed(42)Le programme suivant permet d’étudier l’espérance de M pour différentes valeurs de a et de n (ici a représente les lignes et appartient à [0; 160] tandis que n appartient à [1, 100] et correspond aux colonnes) :
set.seed(42)
a = 160
n = 100
dm <- data.frame(x = 1:a)
for(i in 1:a) {
for(j in 1:n) {
R = replicate(100, max(sample(0:i, j, T)))
dm[i:i, j:j] = mean(R)
}
}
colnames(dm) <- 1:n #Pour nommer les colonnes avec les différentes valeurs prises par n
rownames(dm) <- 1:a #Pour nommer les lignes avec les différentes valeurs prises par a
dm## 1 2 3 4 5 6 7 8 9 10
## 1 0.55 0.65 0.84 0.93 0.98 1.00 0.97 1.00 0.99 1.00
## 2 1.12 1.55 1.71 1.79 1.89 1.91 1.93 1.96 1.99 2.00
## 3 1.71 1.85 2.35 2.66 2.79 2.80 2.90 2.93 2.94 2.96
## 4 1.93 2.85 3.16 3.44 3.60 3.77 3.69 3.83 3.83 3.88
## 5 2.46 3.52 3.98 4.12 4.54 4.59 4.66 4.77 4.74 4.85
## 6 3.08 4.15 4.63 4.90 5.40 5.38 5.65 5.69 5.80 5.69
## 7 3.55 4.94 5.46 5.93 6.15 6.42 6.49 6.47 6.67 6.68
## 8 3.72 5.48 6.26 6.66 6.95 7.09 7.43 7.38 7.40 7.56
## 9 4.10 6.23 6.89 7.51 7.80 8.12 8.13 8.33 8.32 8.45
## 10 4.74 6.45 7.98 8.25 8.43 9.00 8.89 9.18 9.40 9.42
## 11 6.13 7.44 8.48 8.96 9.33 9.87 9.88 10.16 10.09 10.33
## 12 5.66 7.84 9.57 9.86 10.16 10.69 10.94 11.02 11.23 11.16
## 13 6.75 9.23 9.90 10.47 11.29 11.30 11.82 11.95 12.03 12.20
## 14 6.41 10.36 10.33 11.59 12.09 12.55 12.30 12.93 13.12 13.07
## 15 7.24 10.90 11.43 12.23 12.43 13.34 13.61 13.93 14.01 13.89
## 16 8.33 10.58 11.95 12.57 13.58 14.13 14.25 14.55 14.50 15.00
## 17 8.24 11.76 13.01 13.32 13.91 14.96 14.99 15.65 15.52 16.02
## 18 8.39 11.96 13.33 14.95 15.24 15.62 16.58 16.12 16.39 16.79
## 19 10.62 12.91 14.11 15.36 16.08 16.03 17.04 17.26 17.46 17.64
## 20 8.84 13.72 14.51 16.17 17.45 17.51 17.72 18.09 18.40 18.73
## 21 10.66 14.25 16.02 17.33 17.88 18.39 18.80 19.13 19.47 19.13
## 22 11.80 15.34 16.95 17.54 18.82 19.52 19.71 20.20 20.07 20.55
## 23 12.03 15.94 17.63 18.97 19.27 20.54 20.72 20.58 20.68 21.08
## 24 10.93 16.45 17.42 20.20 20.53 20.79 21.32 21.90 22.09 22.41
## 25 12.27 16.88 18.55 20.13 21.00 21.72 21.88 22.35 22.83 23.16
## 26 12.34 18.30 19.73 21.63 22.30 21.88 23.02 23.36 23.47 23.74
## 27 13.98 18.80 20.12 21.68 21.94 23.36 23.88 24.25 24.77 25.13
## 28 12.65 18.47 21.40 21.96 23.91 24.37 24.91 24.90 25.48 25.84
## 29 14.34 18.99 22.83 23.91 24.35 25.33 25.93 25.91 26.42 26.81
## 30 14.71 20.36 22.82 24.59 25.09 26.25 26.60 27.02 27.54 27.59
## 31 15.61 21.62 22.79 25.49 26.02 26.23 27.13 27.51 27.93 28.73
## 32 15.29 23.17 24.28 26.10 26.90 28.40 28.89 28.61 29.01 29.55
## 33 16.78 22.32 25.35 26.80 27.90 28.63 28.76 30.21 29.85 30.32
## 34 16.58 22.31 24.46 27.80 28.53 30.12 29.20 31.52 31.08 31.50
## 35 18.82 24.47 26.67 28.84 29.88 29.36 31.56 31.69 31.41 32.17
## 36 17.72 23.34 27.51 29.12 30.48 31.96 31.77 31.80 32.47 33.18
## 37 16.86 26.68 27.12 30.12 31.47 32.49 32.51 32.94 33.76 34.10
## 38 17.33 25.67 29.06 30.79 32.34 32.93 33.05 33.65 34.53 35.20
## 39 20.32 25.28 29.00 30.58 32.87 34.59 34.75 35.13 35.31 35.97
## 40 20.89 26.02 29.92 32.08 33.63 35.78 35.77 36.56 36.45 36.89
## 41 18.10 28.14 31.54 32.89 34.31 36.16 36.14 36.96 37.29 37.83
## 42 20.69 27.93 33.18 33.65 36.48 36.25 37.34 37.45 37.80 38.70
## 43 19.97 28.20 31.84 34.68 35.77 36.52 38.10 38.87 39.02 39.96
## 44 23.13 31.97 33.08 35.21 35.50 38.68 38.54 39.12 40.69 40.09
## 45 22.52 28.57 34.36 35.78 37.67 39.81 39.78 40.08 41.24 41.15
## 46 22.56 29.73 36.28 37.40 37.88 40.16 40.09 41.49 41.89 41.65
## 47 24.24 31.11 36.26 37.41 39.84 41.25 41.59 41.75 42.73 43.29
## 48 22.27 31.62 36.88 38.83 40.94 41.54 42.14 43.44 43.49 44.71
## 49 24.74 33.48 35.76 38.33 39.49 41.66 42.57 43.22 44.57 44.58
## 50 25.84 34.04 37.90 40.53 42.43 43.29 44.63 45.70 45.39 46.52
## 51 26.69 34.76 39.81 41.39 41.85 44.12 45.38 44.60 46.38 46.62
## 52 24.79 34.32 38.08 41.55 43.81 45.65 46.20 46.92 46.65 47.94
## 53 30.29 34.76 39.30 42.74 43.25 46.84 47.46 46.83 47.59 48.84
## 54 27.37 36.97 38.89 42.71 45.49 47.56 47.15 48.19 48.49 49.47
## 55 28.52 36.53 40.66 43.73 44.50 47.37 47.89 47.86 50.50 49.96
## 56 27.04 35.93 42.38 45.48 45.75 48.47 49.36 51.24 50.09 52.03
## 57 29.76 40.55 43.45 46.64 48.75 48.33 51.16 51.34 51.50 51.46
## 58 30.70 38.79 43.58 45.85 49.17 51.38 51.15 51.33 53.02 53.06
## 59 30.43 38.51 43.65 47.80 49.55 50.55 51.03 51.66 53.28 53.92
## 60 28.08 43.16 45.32 46.60 49.52 52.88 52.76 53.54 53.23 54.93
## 61 30.38 42.52 43.95 48.91 49.64 51.96 53.24 54.83 55.24 55.61
## 62 28.32 41.45 47.31 50.70 51.45 53.55 55.83 55.43 55.43 56.68
## 63 31.70 40.80 45.31 52.66 53.23 54.27 56.39 55.61 56.97 58.09
## 64 33.56 42.83 47.76 50.34 53.36 57.22 54.99 57.13 56.87 58.66
## 65 35.15 45.06 51.71 52.28 54.77 55.05 58.06 57.36 59.56 59.75
## 66 26.97 43.81 48.18 52.93 54.50 57.37 58.35 58.30 60.36 60.11
## 67 33.56 42.16 50.65 54.65 54.96 58.13 58.32 59.65 61.43 61.72
## 68 36.89 43.22 52.02 56.75 57.87 58.18 58.89 60.01 61.54 62.37
## 69 34.91 48.10 52.71 54.51 55.62 60.25 61.34 61.30 62.15 63.67
## 70 37.45 44.99 51.99 55.45 58.33 60.35 61.13 62.50 64.59 63.75
## 71 30.94 46.90 53.59 56.77 58.76 61.11 62.51 63.87 64.38 65.41
## 72 35.42 44.85 54.40 57.49 61.29 60.92 63.46 64.04 64.71 65.44
## 73 34.17 51.86 54.51 56.31 60.80 63.44 63.91 63.79 65.73 67.13
## 74 37.14 49.29 56.43 59.08 62.39 64.50 65.47 67.06 66.32 67.46
## 75 36.97 51.94 55.38 61.65 62.37 64.92 67.37 66.28 67.34 68.99
## 76 39.59 51.15 57.98 61.95 64.71 65.44 67.45 67.27 69.05 68.97
## 77 39.21 53.84 62.59 60.66 63.70 66.73 67.18 68.74 70.53 69.75
## 78 43.01 53.75 56.57 63.91 64.58 67.90 68.30 70.57 69.43 71.93
## 79 39.60 54.33 56.33 61.33 65.17 68.75 70.90 70.13 73.16 71.30
## 80 39.83 53.33 59.98 62.60 66.71 67.34 70.26 72.05 72.17 72.59
## 81 37.71 53.50 60.89 65.94 68.76 70.46 71.47 73.32 73.02 73.01
## 82 38.76 54.26 63.02 64.74 69.40 70.27 72.69 70.58 74.51 74.66
## 83 44.76 54.41 60.81 65.17 70.37 71.47 72.19 75.11 75.48 75.82
## 84 41.94 58.25 62.98 68.30 70.20 71.28 72.71 74.06 76.23 76.39
## 85 40.50 58.39 65.92 67.99 70.50 72.87 73.51 75.04 77.04 78.21
## 86 38.33 58.15 64.78 69.61 70.55 74.30 75.95 77.15 77.86 77.09
## 87 46.28 56.91 65.70 69.37 73.01 75.93 76.22 79.11 78.64 80.32
## 88 51.12 60.55 66.25 69.64 73.61 75.72 76.51 78.79 78.80 80.10
## 89 45.20 59.38 66.45 72.24 75.01 77.20 78.71 81.62 80.18 81.47
## 90 44.44 60.93 66.04 71.75 76.13 76.51 78.73 81.90 82.13 82.68
## 91 48.19 63.47 64.65 72.16 77.28 76.36 80.25 82.92 81.19 84.88
## 92 41.45 57.30 69.19 73.02 77.42 77.67 83.54 81.15 83.27 82.17
## 93 48.04 65.29 70.81 75.02 75.19 79.29 81.41 82.17 85.07 83.97
## 94 43.87 65.55 69.17 74.61 78.97 79.91 82.25 83.77 83.54 85.37
## 95 52.28 63.41 71.03 74.39 81.90 81.42 83.05 84.88 84.82 86.58
## 96 44.93 67.54 72.45 76.19 81.13 85.82 83.71 84.95 86.15 88.71
## 97 48.37 67.84 75.74 78.73 81.24 84.14 84.89 85.56 86.70 88.63
## 98 51.63 66.57 73.85 76.25 84.11 83.92 85.85 87.86 88.04 87.74
## 99 47.03 67.48 73.58 80.47 80.23 83.27 84.90 90.27 89.23 90.24
## 100 48.07 65.27 76.66 80.90 82.28 86.99 89.77 88.84 91.11 92.50
## 101 55.78 69.46 76.48 81.23 82.71 86.64 88.45 89.72 91.92 91.83
## 102 49.56 74.34 79.26 80.07 87.58 87.42 90.76 90.80 92.69 93.28
## 103 45.12 67.02 77.23 84.08 85.39 88.77 90.27 90.17 91.83 93.10
## 104 50.20 67.80 80.61 83.65 87.85 89.94 91.19 94.40 96.21 94.25
## 105 51.12 66.46 79.24 81.41 84.42 92.62 91.65 94.45 95.99 95.40
## 106 55.70 73.25 79.38 83.25 90.33 91.07 93.36 93.35 94.73 96.46
## 107 47.81 68.83 77.01 86.70 89.65 90.83 94.61 96.04 95.34 98.06
## 108 57.84 69.74 81.22 88.65 90.13 90.31 94.46 96.36 97.54 98.70
## 109 52.45 72.74 82.73 86.72 92.95 90.69 97.96 97.34 98.76 99.05
## 110 55.62 69.04 83.46 91.50 92.36 94.16 96.36 97.17 100.25 99.56
## 111 53.36 77.24 81.20 90.27 92.83 96.32 95.52 100.48 100.24 101.62
## 112 50.92 74.66 85.54 84.07 93.52 97.56 98.91 99.67 100.85 101.98
## 113 52.87 74.77 84.61 92.09 94.48 97.99 98.97 100.57 100.78 104.12
## 114 52.10 74.54 86.11 89.60 95.31 98.72 99.50 100.41 101.10 103.95
## 115 58.74 75.80 87.82 92.48 97.06 96.90 99.75 103.77 105.52 104.85
## 116 55.66 76.55 88.99 90.56 95.46 98.42 101.50 104.63 102.54 106.75
## 117 56.73 84.73 89.58 90.54 100.56 100.07 101.52 103.45 106.14 105.06
## 118 57.99 79.59 90.02 96.13 98.70 99.88 105.19 105.17 106.72 107.72
## 119 61.69 77.13 91.33 95.15 101.14 102.75 103.64 105.90 107.74 109.43
## 120 56.48 81.63 90.16 96.78 103.24 103.75 104.25 107.62 109.42 108.27
## 121 63.05 83.55 88.05 97.42 103.29 106.61 108.68 107.77 109.00 111.01
## 122 60.60 80.27 94.03 95.94 102.14 103.85 106.81 110.47 107.76 110.83
## 123 58.37 79.17 93.43 98.26 103.71 106.99 106.85 108.99 109.25 114.30
## 124 60.03 79.19 99.79 99.43 103.83 104.74 110.33 110.85 110.31 112.41
## 125 67.40 90.25 92.58 101.93 108.12 109.08 106.57 110.93 112.91 112.98
## 126 64.40 87.09 93.99 102.65 106.29 107.08 109.43 112.76 113.11 115.56
## 127 66.16 82.13 96.67 99.50 107.86 110.41 109.18 113.94 114.17 116.45
## 128 59.20 89.52 92.47 99.74 107.34 108.94 114.26 112.34 115.09 116.98
## 129 67.03 87.29 95.49 104.75 108.72 110.18 112.47 114.36 115.84 117.56
## 130 67.31 84.80 94.34 101.25 108.14 112.49 113.92 117.22 117.45 120.28
## 131 64.53 93.27 98.76 107.69 108.66 113.33 115.59 116.65 118.65 117.71
## 132 66.42 87.86 101.47 113.03 111.59 112.95 113.18 115.84 120.90 119.13
## 133 62.87 89.14 98.00 108.37 113.23 114.98 117.01 120.78 120.07 121.11
## 134 65.82 87.96 101.67 105.74 112.26 118.65 117.67 120.16 122.68 122.98
## 135 72.12 84.99 102.17 110.58 114.11 118.15 120.30 122.23 120.61 123.73
## 136 65.35 87.05 101.33 107.55 116.35 119.30 118.20 119.79 122.42 125.03
## 137 65.34 89.89 103.31 111.52 116.97 119.26 121.93 122.48 122.54 125.86
## 138 66.88 89.77 104.26 109.06 113.81 119.10 121.56 122.78 126.88 124.88
## 139 67.38 94.16 99.17 113.60 117.62 117.86 121.84 123.61 125.44 124.65
## 140 73.14 97.21 108.71 115.31 114.70 120.13 121.58 125.08 126.67 127.83
## 141 75.36 92.38 105.32 112.47 121.10 121.05 124.92 127.98 128.56 129.38
## 142 70.83 98.03 105.09 112.91 116.93 121.81 123.93 127.93 128.42 129.45
## 143 69.70 96.75 102.11 116.57 118.87 123.03 125.68 127.90 130.15 129.56
## 144 78.31 95.66 107.30 119.13 121.42 126.33 126.06 129.01 127.82 130.21
## 145 71.94 92.12 110.65 115.95 124.47 122.94 126.49 128.32 132.41 131.19
## 146 72.95 101.35 110.85 117.19 123.37 130.08 127.69 129.25 130.76 135.82
## 147 71.85 99.38 107.38 114.43 124.59 126.56 128.89 129.28 133.11 133.44
## 148 74.33 102.27 116.05 121.69 124.46 128.55 128.81 132.32 133.94 135.56
## 149 76.96 105.13 112.03 120.83 122.97 131.60 130.38 134.30 136.17 135.63
## 150 73.33 103.35 115.33 121.05 129.55 129.43 132.30 133.42 134.96 136.63
## 151 74.40 98.44 116.62 121.32 127.46 129.11 132.19 135.04 136.33 136.78
## 152 75.83 93.00 114.49 122.03 123.71 133.02 130.05 135.09 136.81 138.02
## 153 73.35 103.30 108.44 128.02 128.58 131.98 134.81 134.85 136.91 139.74
## 154 83.51 97.48 114.64 118.64 128.26 131.93 133.95 136.62 138.55 140.66
## 155 78.34 101.48 113.00 123.19 129.10 131.28 133.76 138.52 140.13 140.41
## 156 80.71 104.52 117.34 123.45 128.36 134.77 135.82 138.38 138.20 142.02
## 157 74.28 106.08 117.87 124.90 133.99 132.69 137.64 136.57 141.16 145.37
## 158 78.60 105.37 120.46 127.36 132.80 136.02 141.26 140.38 144.09 145.18
## 159 80.95 103.31 116.50 127.72 132.24 136.46 140.34 141.04 141.19 144.08
## 160 81.38 104.66 123.62 129.55 133.41 136.96 142.21 143.56 143.80 144.90
## 11 12 13 14 15 16 17 18 19 20
## 1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
## 2 1.98 1.97 1.99 2.00 2.00 2.00 2.00 2.00 2.00 2.00
## 3 2.98 2.96 2.99 2.96 3.00 3.00 3.00 2.98 3.00 3.00
## 4 3.92 3.95 3.94 3.96 3.93 3.96 3.99 3.96 3.98 4.00
## 5 4.88 4.88 4.91 4.95 4.91 4.93 4.94 4.93 4.98 4.99
## 6 5.85 5.82 5.92 5.93 5.94 5.92 5.82 5.92 5.96 5.96
## 7 6.76 6.79 6.88 6.85 6.87 6.83 6.87 6.92 6.90 6.94
## 8 7.66 7.72 7.82 7.75 7.89 7.83 7.83 7.82 7.90 7.88
## 9 8.70 8.52 8.73 8.68 8.72 8.81 8.84 8.75 8.83 8.84
## 10 9.49 9.51 9.59 9.76 9.78 9.76 9.82 9.71 9.78 9.76
## 11 10.47 10.54 10.66 10.65 10.78 10.56 10.70 10.76 10.79 10.87
## 12 11.48 11.33 11.49 11.59 11.52 11.69 11.70 11.62 11.73 11.81
## 13 12.37 12.36 12.29 12.52 12.53 12.52 12.70 12.65 12.66 12.71
## 14 13.11 13.31 13.37 13.53 13.51 13.49 13.45 13.51 13.65 13.64
## 15 14.08 14.42 14.29 14.44 14.56 14.44 14.61 14.48 14.59 14.53
## 16 15.11 15.18 15.40 15.44 15.40 15.39 15.51 15.57 15.59 15.48
## 17 15.82 16.00 16.15 16.21 16.33 16.53 16.40 16.53 16.52 16.57
## 18 16.93 17.10 17.19 17.01 17.31 17.08 17.27 17.34 17.45 17.62
## 19 17.48 18.14 18.03 18.06 18.31 18.27 18.32 18.44 18.43 18.58
## 20 18.75 18.97 18.96 18.96 18.96 19.39 19.49 19.29 19.45 19.45
## 21 19.30 19.78 19.63 19.91 20.02 20.28 20.35 20.33 20.25 20.39
## 22 20.56 20.98 20.82 20.87 20.94 20.85 21.34 21.30 21.30 21.36
## 23 21.28 21.46 21.72 21.64 22.03 22.08 22.08 22.00 22.17 22.27
## 24 21.89 22.49 22.40 22.73 22.64 23.01 22.73 23.18 23.33 23.44
## 25 23.44 23.38 23.47 23.74 23.77 23.99 24.15 23.90 24.05 24.23
## 26 24.18 24.64 24.58 24.72 24.94 24.86 25.05 25.09 25.13 25.30
## 27 25.08 24.97 25.13 25.92 25.71 25.56 25.98 25.90 26.11 26.33
## 28 26.21 26.58 26.49 26.80 26.81 26.58 27.19 26.94 26.78 27.11
## 29 27.16 27.19 27.44 27.53 27.48 27.56 27.70 27.86 27.93 28.03
## 30 27.87 28.13 28.26 28.39 28.40 28.09 28.82 28.98 28.82 28.82
## 31 28.90 28.90 29.67 29.14 29.41 29.75 29.68 29.93 30.05 29.90
## 32 29.62 29.89 29.94 30.48 30.23 30.33 30.59 30.40 30.64 31.04
## 33 30.62 30.51 30.94 31.27 31.31 31.50 31.40 31.88 31.86 31.67
## 34 31.91 31.45 32.31 32.31 32.31 32.64 32.56 32.63 32.79 32.70
## 35 32.40 32.62 32.69 32.93 33.30 33.28 33.44 33.82 33.63 33.52
## 36 33.21 33.53 33.96 33.94 34.33 34.43 34.32 34.20 34.71 34.72
## 37 34.06 34.02 34.76 34.72 35.37 34.90 35.51 35.30 35.67 35.51
## 38 35.37 35.39 36.09 36.09 36.01 36.03 36.09 36.73 36.45 36.48
## 39 36.17 36.54 36.39 36.90 37.15 37.54 37.30 37.22 37.19 37.65
## 40 37.09 37.15 37.41 38.02 37.88 38.14 38.21 38.02 38.28 38.57
## 41 37.77 37.96 38.60 38.38 38.94 39.35 39.14 38.98 38.99 39.27
## 42 38.56 39.60 39.32 39.28 40.16 39.67 40.08 40.38 40.08 40.48
## 43 39.38 40.34 40.35 40.76 41.06 41.09 41.09 41.36 41.45 41.44
## 44 40.23 41.17 41.19 41.08 41.51 41.75 42.38 42.27 42.42 42.36
## 45 41.76 41.46 42.22 42.47 42.92 42.83 42.59 43.05 42.93 43.21
## 46 42.23 42.47 43.44 43.25 43.41 43.71 43.97 44.35 43.97 44.36
## 47 43.54 43.29 44.40 44.12 44.25 45.13 44.91 45.15 44.99 45.12
## 48 44.43 44.95 44.54 45.27 45.76 45.34 45.87 45.94 46.02 46.20
## 49 45.59 45.54 45.88 45.86 45.87 46.19 46.66 46.90 46.78 47.24
## 50 46.55 46.41 46.62 47.10 47.17 47.66 47.34 47.83 48.25 47.88
## 51 47.15 46.91 47.90 47.55 48.65 48.07 48.61 49.09 48.88 49.09
## 52 48.29 48.33 48.30 49.00 49.36 49.17 49.89 49.52 49.71 50.02
## 53 48.34 49.19 49.20 50.25 50.04 50.35 50.53 50.68 50.52 51.29
## 54 50.30 50.44 50.59 50.24 50.80 50.97 51.25 51.84 51.87 51.88
## 55 51.08 51.28 51.73 51.92 51.96 52.23 52.86 52.38 52.86 52.94
## 56 51.70 52.20 52.03 52.55 52.31 53.69 52.88 53.21 53.82 53.71
## 57 52.26 52.72 53.16 53.47 53.58 53.84 54.67 54.68 54.55 54.93
## 58 53.68 53.61 54.62 54.23 54.57 54.60 55.23 55.67 55.60 55.02
## 59 54.47 55.20 55.59 55.38 55.83 56.38 55.88 56.69 56.79 56.74
## 60 55.54 55.80 56.43 56.87 56.12 56.73 57.51 57.71 57.67 56.65
## 61 56.29 56.68 57.08 56.73 57.64 58.15 58.01 58.02 57.97 58.86
## 62 57.60 58.14 58.33 58.42 59.18 58.94 59.16 59.35 59.42 59.37
## 63 57.68 58.11 59.50 58.91 60.32 60.28 59.81 60.30 60.81 60.62
## 64 58.51 60.39 60.09 60.08 60.25 60.69 60.84 60.94 61.25 61.64
## 65 59.79 59.91 61.24 61.05 61.17 61.36 61.41 61.96 62.64 61.75
## 66 60.37 61.46 61.81 62.23 62.21 62.20 62.79 62.92 63.11 63.46
## 67 62.25 61.88 62.46 62.15 63.61 63.80 63.74 64.07 64.02 64.50
## 68 62.66 62.64 63.46 63.90 64.60 63.98 64.96 65.28 64.81 65.18
## 69 63.99 64.56 65.13 64.62 65.18 64.59 65.96 66.01 66.08 65.71
## 70 64.69 64.53 65.36 65.35 66.10 66.59 66.52 67.13 66.91 67.37
## 71 65.13 65.93 66.91 66.72 67.07 67.01 67.68 67.73 68.19 68.15
## 72 67.09 66.23 67.61 67.17 67.24 68.34 68.35 68.92 68.67 69.28
## 73 67.83 68.21 68.10 68.52 68.27 68.56 69.82 69.68 69.75 70.08
## 74 68.73 69.13 68.48 69.50 70.30 70.63 70.24 70.59 70.88 70.66
## 75 69.51 70.19 69.60 71.04 70.94 71.33 71.31 71.46 71.87 71.89
## 76 70.50 70.67 71.46 71.54 71.39 71.84 72.43 72.33 72.59 73.15
## 77 71.66 70.49 72.77 71.77 72.64 72.96 73.15 73.88 73.52 73.79
## 78 72.44 73.09 73.25 73.17 73.29 74.15 73.36 73.97 74.47 74.27
## 79 73.28 73.09 72.48 73.53 74.18 74.85 74.84 74.89 75.09 76.38
## 80 72.35 74.53 73.65 75.15 75.14 75.71 76.27 76.26 76.22 75.83
## 81 75.76 75.45 75.02 75.85 75.87 75.77 76.47 76.65 77.17 77.79
## 82 75.62 76.23 76.61 77.63 78.12 77.37 77.58 78.87 78.62 77.44
## 83 75.69 76.56 76.59 78.02 79.16 77.73 79.04 79.75 79.83 80.03
## 84 77.61 77.08 78.07 79.42 78.09 79.03 80.37 80.06 79.86 80.84
## 85 77.62 78.86 79.64 80.02 79.78 80.89 80.74 80.74 81.54 80.31
## 86 79.34 80.06 81.02 80.99 81.13 81.36 82.72 81.67 81.95 82.23
## 87 79.11 81.06 81.45 80.77 82.07 82.13 83.67 82.50 83.37 82.95
## 88 81.86 81.10 82.65 81.56 83.35 83.99 84.06 84.06 84.44 84.31
## 89 82.46 83.16 84.19 83.71 84.80 84.34 84.78 84.43 84.84 85.04
## 90 83.06 83.49 83.24 84.54 85.37 85.38 85.55 85.45 86.11 86.40
## 91 83.75 84.19 85.17 85.33 85.70 86.28 86.93 87.09 85.81 87.20
## 92 84.41 85.33 85.01 85.98 86.44 87.25 88.63 87.67 87.48 88.39
## 93 86.26 85.44 87.31 86.84 87.28 88.02 89.07 89.03 88.29 89.19
## 94 86.46 87.90 88.31 87.28 89.83 89.21 88.76 89.48 89.55 90.36
## 95 88.28 88.80 89.80 89.28 89.95 90.36 90.46 90.01 91.07 90.00
## 96 87.70 89.62 89.41 90.56 89.88 90.87 90.50 90.65 91.80 91.44
## 97 89.59 90.48 90.45 91.17 91.08 91.26 92.34 91.74 92.84 92.64
## 98 90.12 91.83 90.71 91.49 93.17 92.32 93.42 93.71 93.16 93.83
## 99 91.73 92.83 91.94 92.71 93.74 92.79 94.49 93.98 93.73 95.47
## 100 91.47 92.64 93.53 93.86 93.65 95.09 94.00 95.12 95.12 95.58
## 101 93.11 94.51 93.64 95.38 94.48 95.45 95.63 95.64 96.09 97.04
## 102 92.35 94.12 95.77 94.86 96.22 96.53 97.32 97.42 97.77 97.49
## 103 94.43 94.78 95.46 97.20 97.29 96.84 96.86 97.99 97.68 98.71
## 104 97.05 96.26 96.69 96.69 97.37 98.93 97.73 98.85 99.40 99.12
## 105 96.23 98.13 97.74 97.95 99.60 98.57 98.95 99.62 100.05 99.90
## 106 99.02 98.95 99.16 100.27 99.03 99.11 100.11 101.11 100.20 101.63
## 107 98.22 98.69 100.11 99.88 101.48 100.95 101.44 101.83 103.11 102.72
## 108 97.81 98.91 100.98 100.45 102.04 101.77 102.67 102.55 102.93 103.06
## 109 98.25 100.86 101.35 102.78 102.19 104.21 103.26 102.87 104.20 104.21
## 110 102.37 102.04 102.80 102.31 102.20 103.72 104.51 104.68 105.84 104.19
## 111 101.86 104.04 103.57 104.02 104.49 105.66 105.88 104.86 106.60 106.00
## 112 104.08 103.00 105.02 105.24 105.10 106.23 106.38 106.60 106.66 106.42
## 113 102.81 104.00 105.12 107.04 107.25 107.08 105.34 107.58 108.68 108.03
## 114 104.93 105.65 106.91 107.28 107.16 106.98 107.57 108.50 108.61 109.85
## 115 104.92 106.74 108.10 108.53 107.84 108.12 108.70 109.78 109.65 109.98
## 116 107.25 108.30 109.09 109.27 108.85 110.53 110.66 109.56 110.61 110.02
## 117 107.16 108.72 107.66 109.84 109.45 111.43 111.10 111.55 111.10 110.76
## 118 107.63 109.03 111.24 110.77 110.90 110.48 111.56 112.56 111.68 113.15
## 119 111.13 110.16 111.13 112.80 111.86 113.48 112.93 113.50 113.50 113.49
## 120 109.66 111.19 112.26 112.46 113.50 113.12 113.48 113.74 115.25 114.94
## 121 110.06 111.49 111.60 113.25 113.65 115.43 115.30 114.69 115.69 116.09
## 122 112.16 112.09 113.80 113.73 114.32 114.96 115.83 116.01 116.82 116.27
## 123 113.74 114.64 116.03 114.74 115.79 116.67 117.14 117.34 117.31 117.14
## 124 115.05 114.10 115.15 117.28 117.14 117.34 117.13 118.25 117.78 118.32
## 125 113.80 116.55 117.00 117.87 118.28 117.31 117.72 119.15 118.97 119.30
## 126 116.47 115.99 117.56 117.25 117.45 120.21 119.00 119.43 119.63 120.37
## 127 117.33 117.76 118.95 118.66 119.19 119.70 120.49 120.75 120.77 121.39
## 128 116.65 118.11 119.86 118.89 120.19 120.86 121.50 122.05 122.22 121.70
## 129 118.36 120.82 120.52 120.96 119.03 121.93 121.65 122.06 123.11 122.49
## 130 119.11 120.33 120.92 122.21 121.78 122.42 123.80 122.53 123.07 124.51
## 131 119.84 120.67 122.17 122.06 123.24 123.80 124.07 124.10 125.42 124.48
## 132 121.88 122.98 122.11 123.72 124.18 124.88 125.65 125.34 125.20 125.47
## 133 122.04 122.28 123.42 124.92 125.95 125.61 126.22 126.24 126.64 126.92
## 134 122.35 125.38 126.89 124.15 125.46 124.88 126.26 128.14 128.07 128.65
## 135 124.46 125.31 126.15 125.43 126.62 127.91 126.94 128.52 130.12 129.20
## 136 123.55 125.82 127.64 125.63 126.92 128.87 129.88 129.18 130.34 130.87
## 137 124.06 125.71 126.63 129.27 128.85 128.81 128.77 131.43 130.93 131.18
## 138 127.63 127.64 128.60 130.53 128.21 130.65 132.24 130.38 131.72 131.86
## 139 126.31 127.40 130.83 130.07 130.90 131.32 131.11 132.42 133.10 133.24
## 140 127.93 131.54 131.20 129.98 131.19 132.26 132.87 132.94 132.74 133.55
## 141 129.33 129.52 132.01 131.63 133.06 133.18 132.89 134.69 135.01 135.20
## 142 131.50 131.26 133.13 133.24 134.40 134.09 134.52 134.72 136.30 135.24
## 143 130.48 131.65 133.06 133.92 134.07 135.62 135.39 135.41 136.05 137.10
## 144 132.82 133.24 133.68 134.42 136.10 135.84 137.53 135.51 136.49 138.28
## 145 132.37 133.37 135.39 135.83 136.31 136.26 137.68 137.75 138.23 138.71
## 146 134.32 135.01 135.91 137.96 136.15 136.92 140.38 138.24 138.66 138.18
## 147 136.38 136.91 135.60 137.61 138.52 138.11 139.11 141.28 140.09 140.92
## 148 135.50 136.50 134.57 138.65 139.13 139.36 139.12 140.66 140.97 141.03
## 149 135.75 138.16 138.44 139.44 140.15 138.33 140.52 140.96 142.60 142.73
## 150 138.87 138.16 139.58 141.00 142.58 140.89 141.75 143.25 143.04 142.83
## 151 138.55 140.04 141.82 140.02 142.15 143.08 144.15 142.65 144.07 143.63
## 152 139.72 140.51 141.21 143.24 142.71 143.00 143.25 142.94 144.15 145.98
## 153 140.29 140.78 142.21 142.44 144.97 144.00 145.81 146.28 145.62 146.33
## 154 142.49 144.64 143.76 142.99 145.01 144.92 145.34 146.35 148.22 146.70
## 155 143.07 144.30 144.00 143.69 144.48 145.56 147.10 147.59 147.31 147.79
## 156 144.07 144.25 146.55 145.17 145.92 146.36 147.07 147.13 149.85 149.96
## 157 144.93 145.04 146.74 147.76 147.35 148.53 148.27 149.15 150.78 148.95
## 158 147.62 146.15 146.87 146.87 149.97 148.98 151.40 148.85 150.42 150.02
## 159 146.36 146.61 146.73 150.62 148.49 150.39 150.97 150.71 152.18 150.93
## 160 147.16 148.69 149.08 147.95 151.06 151.74 152.21 151.99 152.40 153.52
## 21 22 23 24 25 26 27 28 29 30
## 1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
## 2 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
## 3 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
## 4 3.99 3.98 3.99 4.00 4.00 4.00 4.00 4.00 4.00 4.00
## 5 5.00 4.99 4.99 4.97 4.98 4.99 4.99 5.00 4.99 5.00
## 6 5.95 5.95 5.97 5.99 5.98 6.00 6.00 5.97 5.99 5.99
## 7 6.97 6.93 6.94 6.97 6.99 6.97 6.99 7.00 6.94 7.00
## 8 7.91 7.94 7.92 7.98 7.93 7.97 7.95 7.94 7.97 7.96
## 9 8.82 8.91 8.92 8.90 8.89 8.93 8.91 8.95 8.96 8.93
## 10 9.83 9.86 9.79 9.84 9.94 9.93 9.89 9.87 9.95 9.94
## 11 10.83 10.83 10.88 10.85 10.90 10.86 10.86 10.93 10.93 10.94
## 12 11.73 11.87 11.86 11.83 11.87 11.87 11.91 11.91 11.89 11.93
## 13 12.73 12.83 12.80 12.78 12.84 12.83 12.82 12.90 12.79 12.89
## 14 13.70 13.75 13.66 13.82 13.79 13.79 13.80 13.81 13.83 13.89
## 15 14.60 14.77 14.65 14.84 14.80 14.75 14.85 14.82 14.84 14.86
## 16 15.73 15.70 15.67 15.76 15.70 15.80 15.82 15.77 15.76 15.78
## 17 16.49 16.64 16.55 16.66 16.75 16.74 16.86 16.85 16.81 16.77
## 18 17.46 17.53 17.70 17.70 17.77 17.72 17.73 17.71 17.80 17.75
## 19 18.56 18.47 18.65 18.60 18.68 18.57 18.68 18.74 18.76 18.78
## 20 19.41 19.32 19.44 19.54 19.66 19.62 19.59 19.66 19.80 19.61
## 21 20.28 20.48 20.35 20.55 20.63 20.63 20.64 20.57 20.64 20.63
## 22 21.43 21.35 21.48 21.41 21.46 21.48 21.60 21.57 21.70 21.57
## 23 22.46 22.45 22.29 22.48 22.48 22.59 22.59 22.51 22.72 22.46
## 24 23.19 23.35 23.05 23.52 23.49 23.51 23.45 23.52 23.44 23.56
## 25 24.14 24.09 24.47 24.47 24.35 24.29 24.48 24.47 24.70 24.53
## 26 25.36 25.07 25.32 25.37 25.48 25.37 25.44 25.35 25.56 25.62
## 27 26.20 26.04 26.07 26.23 26.28 26.40 26.46 26.45 26.54 26.67
## 28 27.08 27.17 27.32 27.28 27.24 27.45 27.43 27.45 27.46 27.41
## 29 28.16 27.99 28.18 28.11 28.23 28.31 28.35 28.28 28.47 28.47
## 30 28.70 28.96 29.01 29.19 29.22 29.27 29.45 29.30 29.41 29.48
## 31 30.10 30.12 30.07 30.24 30.20 30.16 30.24 30.21 30.55 30.41
## 32 31.04 31.04 31.01 30.96 30.92 31.21 31.18 31.56 31.43 31.29
## 33 31.91 31.88 32.09 32.03 31.96 32.10 32.07 32.31 32.21 32.42
## 34 32.87 32.67 33.11 32.73 33.04 33.39 33.01 33.25 33.35 33.18
## 35 33.77 33.81 33.77 34.06 34.06 34.10 34.14 34.26 34.26 34.32
## 36 34.55 34.74 34.84 34.94 35.02 34.71 35.13 35.24 35.11 35.33
## 37 35.83 35.91 36.04 35.72 36.19 35.89 35.90 36.22 36.21 36.28
## 38 36.37 36.95 36.88 36.89 36.76 36.80 37.21 37.09 37.34 37.09
## 39 37.53 37.35 37.91 37.92 37.86 37.70 38.32 38.14 37.97 38.17
## 40 38.38 38.80 38.64 38.84 39.22 38.86 38.76 39.02 38.83 39.05
## 41 39.96 39.55 39.67 39.79 39.66 40.08 39.98 40.13 40.21 40.13
## 42 40.27 40.68 40.74 40.46 40.82 40.78 40.98 41.24 41.11 40.82
## 43 41.39 41.46 41.37 41.43 41.79 41.69 41.94 42.12 41.92 41.70
## 44 42.51 42.18 42.56 42.80 42.75 42.83 42.92 42.70 42.86 43.04
## 45 43.55 43.81 43.65 43.48 43.62 43.84 43.59 43.94 43.71 43.83
## 46 44.40 44.34 44.46 44.42 44.55 45.10 44.95 44.99 44.83 44.80
## 47 45.01 45.23 45.49 45.81 45.33 45.61 45.55 45.89 45.92 46.01
## 48 46.42 46.31 46.63 46.41 46.49 46.77 46.75 46.65 46.90 46.99
## 49 47.48 46.81 47.29 47.77 47.49 47.16 47.69 48.00 47.94 47.93
## 50 48.14 48.06 48.57 48.22 48.44 48.51 48.82 48.69 48.72 48.70
## 51 49.22 49.48 49.51 49.39 49.66 49.06 49.49 49.86 49.62 49.58
## 52 50.18 50.19 50.47 50.39 50.43 50.49 50.72 50.60 50.57 50.91
## 53 50.97 50.99 51.21 50.93 51.46 51.28 51.42 51.82 51.69 51.60
## 54 51.93 52.17 51.64 52.60 52.23 52.77 52.51 52.52 52.46 52.35
## 55 53.02 53.10 53.00 53.30 53.29 52.77 53.59 53.19 53.46 53.45
## 56 53.82 54.13 54.12 53.71 54.85 54.11 54.05 54.46 54.55 54.54
## 57 54.31 55.20 55.29 55.10 55.23 55.54 55.13 55.60 55.77 56.00
## 58 55.84 56.00 55.60 56.27 56.24 56.17 56.59 56.44 56.55 56.72
## 59 56.48 56.90 57.11 57.38 56.84 56.88 57.25 57.39 57.31 57.93
## 60 58.16 57.11 57.63 58.02 58.04 58.14 58.33 58.26 58.38 58.45
## 61 58.75 58.04 58.80 58.85 59.04 58.96 59.23 59.72 59.24 59.31
## 62 59.68 59.76 59.75 59.76 59.94 59.97 60.14 59.86 60.35 60.60
## 63 60.84 60.58 61.16 60.66 60.75 61.11 61.45 61.03 61.77 61.55
## 64 61.74 61.76 61.88 61.53 61.87 62.42 61.99 62.25 62.41 62.37
## 65 62.42 62.06 62.69 62.98 63.13 62.78 62.93 63.39 63.62 63.32
## 66 63.15 63.91 63.58 63.55 63.92 64.01 64.13 64.26 64.23 64.20
## 67 64.15 64.51 64.87 65.16 64.76 64.68 64.92 65.39 65.05 65.48
## 68 65.38 65.56 65.67 65.52 66.45 65.75 66.01 66.13 66.26 66.18
## 69 66.17 66.57 66.65 67.00 67.03 66.89 67.08 67.40 67.32 67.46
## 70 66.98 67.24 67.70 68.20 67.41 68.16 68.05 67.72 68.17 68.23
## 71 68.34 68.66 68.24 68.82 67.99 68.60 69.15 68.91 68.95 69.25
## 72 69.30 69.09 69.77 69.45 69.81 69.87 69.79 70.09 70.44 70.34
## 73 70.24 70.36 70.78 69.99 70.91 70.52 70.95 71.37 70.87 71.01
## 74 71.10 70.80 71.24 71.18 71.78 71.70 71.71 72.10 71.84 71.99
## 75 71.77 72.30 72.18 72.45 72.38 72.47 72.33 72.95 73.39 72.94
## 76 72.46 72.82 73.10 73.17 73.62 73.48 73.46 73.66 74.27 74.12
## 77 74.63 73.93 73.69 74.54 74.35 74.44 74.73 74.72 74.86 75.12
## 78 74.91 75.15 75.11 75.18 75.07 75.34 75.49 75.86 76.25 75.34
## 79 76.03 76.27 76.72 76.24 76.70 76.21 76.87 76.52 76.42 76.32
## 80 77.15 77.05 76.82 77.21 77.60 77.66 77.28 77.87 77.57 77.65
## 81 77.70 78.52 78.45 78.21 77.76 77.88 77.81 78.24 78.49 78.40
## 82 79.07 79.04 79.15 78.81 79.51 79.37 79.73 79.68 79.63 79.54
## 83 79.89 79.75 80.16 80.33 80.36 79.83 80.51 80.92 80.53 80.55
## 84 80.98 80.07 80.81 81.74 80.99 80.81 81.44 81.54 81.59 81.30
## 85 82.18 81.29 81.72 82.35 82.25 82.18 82.33 82.41 82.63 83.18
## 86 82.90 82.54 83.51 83.30 82.99 83.21 83.24 83.49 83.10 83.67
## 87 83.78 83.54 84.02 83.56 83.68 83.69 84.14 84.42 84.63 84.76
## 88 85.03 84.62 84.41 84.86 85.26 85.27 85.47 85.82 85.64 85.47
## 89 85.78 85.44 86.24 86.20 86.54 86.39 86.66 86.13 86.27 86.57
## 90 86.95 87.10 86.23 86.94 86.78 86.99 87.05 87.59 86.94 87.21
## 91 87.63 87.74 87.50 87.91 88.21 87.85 88.37 88.42 88.57 88.35
## 92 87.76 88.88 88.60 89.11 88.99 89.03 89.06 89.44 89.09 88.70
## 93 89.76 89.89 89.41 89.73 89.66 89.61 90.63 90.98 90.13 90.94
## 94 89.41 89.86 90.57 90.23 90.60 91.00 90.66 91.14 91.45 91.54
## 95 90.73 90.75 92.38 92.10 91.83 91.94 92.32 91.23 92.32 92.23
## 96 92.03 92.47 92.83 93.06 93.39 93.53 92.99 93.22 94.03 93.92
## 97 92.89 92.73 92.99 93.41 93.80 94.15 93.56 94.14 94.17 94.25
## 98 93.79 93.88 94.47 94.36 94.68 94.91 94.78 95.24 95.22 95.34
## 99 94.27 94.65 95.62 95.63 96.12 95.67 96.26 95.56 95.81 96.13
## 100 97.27 96.27 95.76 96.56 97.00 96.62 97.07 97.43 97.08 97.02
## 101 97.17 95.91 97.24 96.48 97.81 97.81 97.88 97.66 98.33 97.81
## 102 97.17 98.55 97.29 99.07 98.36 98.85 98.02 99.20 98.60 98.62
## 103 98.44 99.23 98.49 99.10 99.48 98.80 100.03 100.70 100.14 100.48
## 104 99.41 99.73 99.92 100.76 100.56 100.81 101.32 101.18 100.89 100.94
## 105 100.05 101.38 100.31 101.61 101.35 102.14 102.00 101.17 102.01 101.77
## 106 101.36 101.24 101.89 101.63 102.67 102.14 102.69 102.93 103.35 102.96
## 107 102.78 102.46 102.21 102.88 103.10 103.16 104.25 103.32 103.77 104.51
## 108 103.54 103.04 103.95 104.16 104.52 104.29 105.04 104.92 104.83 105.63
## 109 104.48 104.14 105.32 105.23 104.97 105.90 105.55 105.66 105.83 106.22
## 110 105.20 104.93 106.44 106.39 105.81 106.55 106.96 106.69 106.75 106.99
## 111 105.92 106.08 106.86 107.07 106.96 107.30 106.48 107.33 107.62 108.56
## 112 107.85 107.55 107.90 108.04 107.68 108.61 107.84 108.47 108.80 108.64
## 113 108.77 108.05 108.43 109.21 109.16 109.44 109.25 110.16 109.81 109.49
## 114 109.51 109.66 109.51 109.74 110.81 110.01 110.52 110.48 110.77 111.18
## 115 110.56 110.54 110.41 110.94 111.48 111.61 111.39 111.50 111.51 111.86
## 116 111.19 111.71 111.40 112.35 112.44 111.04 112.48 113.01 112.89 113.00
## 117 111.80 111.96 112.79 113.12 112.76 113.29 113.28 113.26 113.16 113.64
## 118 113.09 113.62 113.00 114.01 113.52 114.38 114.52 115.00 114.53 114.91
## 119 114.57 114.57 114.33 115.22 114.88 114.60 115.26 115.03 115.54 115.65
## 120 114.57 114.68 115.39 115.94 116.07 115.70 116.26 116.18 115.47 116.64
## 121 116.96 116.49 116.84 117.00 116.84 117.19 117.44 116.99 117.39 117.75
## 122 116.02 117.23 117.67 117.73 117.76 118.11 118.11 118.31 117.59 118.67
## 123 117.98 118.77 117.68 118.76 118.71 118.89 119.57 118.44 120.19 118.83
## 124 119.72 119.39 119.04 119.52 119.56 119.78 119.71 119.93 120.50 120.47
## 125 120.01 118.94 120.20 120.84 120.63 121.01 120.77 121.36 120.75 121.18
## 126 120.01 120.69 120.31 120.83 122.53 121.41 122.16 121.51 122.67 121.68
## 127 121.50 121.93 122.43 121.36 123.10 122.54 122.90 123.21 123.56 123.72
## 128 123.43 122.53 123.47 123.59 124.27 123.54 124.65 123.19 123.67 124.04
## 129 124.26 123.10 124.62 124.71 124.68 125.25 123.86 124.24 124.92 124.81
## 130 124.52 125.36 125.34 125.23 126.12 125.79 126.24 125.50 126.13 125.42
## 131 125.33 125.39 126.48 126.86 126.52 126.84 127.63 126.70 127.60 127.94
## 132 126.00 126.48 127.82 126.66 127.30 127.62 127.37 127.39 128.64 128.23
## 133 127.60 127.03 128.05 127.85 127.49 128.31 128.07 129.11 128.81 129.30
## 134 128.11 127.12 129.63 129.46 129.14 129.41 129.81 130.10 129.25 130.15
## 135 128.95 129.45 130.46 129.46 130.10 129.51 130.27 131.48 130.41 130.93
## 136 129.85 130.74 130.94 130.33 131.47 131.30 131.65 131.86 131.36 131.70
## 137 131.36 130.49 131.11 132.24 132.78 132.57 131.47 132.55 133.45 134.05
## 138 132.13 132.65 132.44 131.84 133.49 133.25 134.27 133.44 133.42 134.09
## 139 133.21 134.06 132.37 133.71 134.10 133.42 134.74 135.32 134.83 135.02
## 140 134.36 134.22 135.23 134.58 134.03 135.47 135.79 135.31 135.81 136.31
## 141 135.30 136.22 136.03 135.82 136.93 135.92 135.68 135.45 136.90 136.84
## 142 135.36 135.94 136.66 136.44 137.20 138.08 137.43 137.49 137.89 137.98
## 143 137.25 136.79 137.08 137.93 138.53 138.15 138.87 138.59 139.13 139.28
## 144 138.54 137.68 138.45 138.74 139.71 138.19 138.94 139.38 139.28 140.15
## 145 140.11 139.10 139.19 138.70 140.27 140.60 139.77 140.86 140.01 140.12
## 146 139.67 140.01 139.41 141.10 140.30 140.88 140.72 141.19 141.98 140.21
## 147 139.54 141.00 141.66 142.18 142.28 141.84 141.52 141.40 143.14 142.41
## 148 141.26 142.21 142.21 142.44 143.08 143.57 143.36 144.52 143.10 143.83
## 149 142.60 143.12 142.31 141.95 142.97 143.44 144.42 143.62 144.44 144.43
## 150 142.89 143.35 144.39 144.81 145.05 143.79 145.31 145.35 145.69 146.24
## 151 144.83 145.84 145.50 145.56 144.99 145.69 146.25 146.16 147.07 147.03
## 152 145.47 145.81 146.93 146.24 146.97 146.70 147.46 148.11 146.88 147.40
## 153 146.85 146.63 147.28 147.71 146.83 147.62 147.55 147.78 148.46 147.94
## 154 146.96 148.19 148.52 148.57 146.46 149.24 148.96 149.35 148.96 149.38
## 155 148.27 148.31 149.99 149.62 149.01 150.20 150.23 150.08 150.50 150.84
## 156 149.24 150.10 149.52 149.39 150.58 150.56 150.99 150.99 150.14 151.12
## 157 151.66 151.01 150.61 151.01 152.13 151.24 152.86 152.56 152.69 152.24
## 158 149.77 152.55 151.87 153.25 152.71 153.07 151.82 153.22 153.38 152.99
## 159 152.59 152.76 151.66 151.19 153.61 152.76 154.00 153.81 154.57 154.17
## 160 152.69 152.76 153.24 153.88 153.13 153.50 154.66 154.82 154.94 155.76
## 31 32 33 34 35 36 37 38 39 40
## 1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
## 2 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
## 3 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
## 4 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00
## 5 4.99 5.00 5.00 5.00 5.00 4.99 5.00 5.00 5.00 5.00
## 6 5.98 5.99 6.00 5.99 6.00 5.99 6.00 6.00 6.00 5.99
## 7 6.97 6.96 6.97 7.00 6.99 7.00 6.99 7.00 6.99 7.00
## 8 7.98 7.97 7.98 7.98 7.98 7.97 7.98 7.99 8.00 8.00
## 9 8.98 8.94 8.96 8.98 8.95 8.99 8.97 8.99 8.99 8.98
## 10 9.97 9.96 9.93 9.98 9.95 9.96 9.98 9.97 9.97 9.98
## 11 10.95 10.91 10.93 10.97 10.98 10.94 10.97 10.95 10.96 10.97
## 12 11.93 11.90 11.96 11.91 11.94 11.89 11.96 11.98 11.94 11.99
## 13 12.83 12.94 12.93 12.89 12.93 12.92 12.94 12.91 12.95 12.92
## 14 13.82 13.80 13.93 13.86 13.92 13.96 13.88 13.95 13.93 13.92
## 15 14.85 14.85 14.87 14.89 14.87 14.84 14.94 14.92 14.97 14.93
## 16 15.82 15.85 15.79 15.86 15.86 15.80 15.89 15.90 15.82 15.97
## 17 16.77 16.80 16.86 16.88 16.82 16.86 16.89 16.83 16.92 16.88
## 18 17.66 17.76 17.88 17.83 17.86 17.83 17.80 17.88 17.88 17.83
## 19 18.74 18.74 18.77 18.69 18.80 18.84 18.80 18.85 18.84 18.86
## 20 19.60 19.73 19.72 19.75 19.82 19.83 19.79 19.81 19.89 19.85
## 21 20.69 20.57 20.78 20.76 20.79 20.73 20.80 20.79 20.83 20.79
## 22 21.70 21.58 21.60 21.86 21.69 21.85 21.82 21.65 21.76 21.70
## 23 22.70 22.65 22.74 22.71 22.71 22.65 22.80 22.70 22.75 22.86
## 24 23.58 23.64 23.54 23.67 23.83 23.74 23.62 23.61 23.70 23.77
## 25 24.55 24.55 24.63 24.50 24.67 24.72 24.71 24.68 24.72 24.82
## 26 25.65 25.71 25.53 25.60 25.64 25.66 25.69 25.67 25.67 25.70
## 27 26.47 26.52 26.65 26.60 26.60 26.75 26.64 26.56 26.70 26.76
## 28 27.49 27.40 27.44 27.70 27.61 27.78 27.68 27.67 27.66 27.59
## 29 28.54 28.52 28.47 28.61 28.63 28.56 28.56 28.52 28.50 28.61
## 30 29.57 29.49 29.56 29.48 29.40 29.54 29.61 29.59 29.46 29.62
## 31 30.35 30.36 30.48 30.33 30.69 30.65 30.69 30.57 30.55 30.63
## 32 31.42 31.48 31.57 31.32 31.51 31.51 31.53 31.58 31.59 31.63
## 33 32.49 32.62 32.43 32.35 32.61 32.38 32.72 32.59 32.52 32.63
## 34 33.23 33.28 33.20 33.39 33.50 33.20 33.52 33.50 33.52 33.61
## 35 34.50 34.23 34.40 34.57 34.39 34.46 34.46 34.60 34.44 34.47
## 36 35.30 35.38 35.31 35.38 35.42 35.51 35.55 35.45 35.46 35.49
## 37 36.34 36.20 36.31 36.35 36.25 36.40 36.30 36.49 36.42 36.49
## 38 37.15 37.31 37.36 37.18 37.43 37.22 37.36 37.50 37.45 37.44
## 39 38.01 38.29 38.05 38.23 38.40 38.38 38.34 38.39 38.52 38.53
## 40 39.25 39.34 39.28 39.26 39.15 39.17 39.29 39.46 39.23 39.14
## 41 40.22 40.33 40.05 40.38 40.46 40.44 40.31 40.44 40.39 40.39
## 42 40.99 41.17 41.15 41.17 41.08 41.27 41.28 41.36 41.42 41.32
## 43 42.18 42.16 42.03 42.30 42.05 42.29 42.12 42.28 42.33 42.49
## 44 42.84 43.19 43.25 43.13 42.99 43.07 43.22 43.25 43.41 43.12
## 45 44.09 44.23 44.25 44.17 44.27 44.23 44.25 44.44 44.10 44.37
## 46 44.85 45.02 45.00 45.02 44.88 45.17 45.40 45.06 45.24 45.28
## 47 45.76 45.99 46.28 46.13 46.11 46.24 46.28 46.29 46.43 46.20
## 48 46.63 47.01 47.07 46.93 46.82 46.87 47.14 47.10 47.39 47.35
## 49 47.93 47.94 48.06 48.13 48.02 48.14 48.03 48.28 48.12 48.22
## 50 48.68 48.90 48.76 49.02 49.14 48.99 48.79 49.31 49.29 49.25
## 51 49.70 49.80 49.86 49.72 49.91 49.93 49.94 50.19 50.13 50.08
## 52 51.08 50.91 51.02 50.83 50.94 50.87 51.05 51.22 51.20 51.16
## 53 51.70 51.86 51.77 51.80 51.80 51.92 52.16 52.06 52.22 52.17
## 54 52.93 52.95 52.90 53.08 53.08 52.94 52.96 53.03 53.05 53.15
## 55 53.89 53.48 53.85 53.81 53.99 54.03 53.84 53.93 54.14 53.96
## 56 54.88 54.54 54.78 54.73 54.79 54.60 54.88 55.05 55.15 54.97
## 57 55.66 55.63 55.70 55.54 55.64 55.82 55.81 56.09 55.76 56.28
## 58 56.87 56.66 56.66 56.67 56.55 56.82 56.83 56.77 57.14 57.08
## 59 57.75 57.29 57.52 57.73 57.77 57.82 57.89 57.49 57.94 58.09
## 60 58.38 58.52 58.71 58.80 58.75 58.76 58.97 59.04 58.99 59.12
## 61 59.59 59.40 59.67 59.77 59.85 59.85 59.55 60.05 59.99 60.08
## 62 60.29 60.47 60.71 60.69 60.52 60.73 61.03 60.93 60.96 60.92
## 63 61.75 61.37 61.48 61.62 61.61 61.75 61.69 62.01 62.05 61.95
## 64 62.32 62.42 62.44 62.62 62.97 62.81 62.74 62.35 62.70 62.96
## 65 63.47 63.36 63.80 63.33 63.63 63.77 63.40 63.71 63.80 63.82
## 66 64.60 64.56 64.33 64.63 64.32 64.57 64.89 64.41 64.59 65.00
## 67 65.15 65.75 65.66 65.67 65.75 65.55 65.53 65.66 65.55 65.96
## 68 66.46 66.56 66.73 66.77 66.74 66.76 66.85 66.66 66.74 66.87
## 69 66.75 67.29 67.04 67.67 67.53 67.78 67.57 67.72 67.72 67.57
## 70 68.08 68.33 68.70 68.30 68.29 68.51 68.67 68.64 68.31 68.90
## 71 69.30 69.30 69.30 69.13 69.30 69.44 69.64 69.78 69.62 69.82
## 72 70.34 70.37 70.24 70.45 70.40 70.64 70.59 70.49 70.62 70.72
## 73 71.07 71.12 71.34 71.18 71.52 71.18 71.69 71.43 71.72 71.67
## 74 72.13 72.35 72.25 72.11 72.05 72.35 72.26 72.09 72.74 72.60
## 75 73.14 73.37 72.77 73.31 73.05 73.33 73.53 73.53 73.56 73.57
## 76 74.37 74.27 74.14 74.28 73.99 74.28 74.40 74.46 74.79 74.58
## 77 75.17 74.91 75.27 75.26 75.29 75.33 75.49 75.10 75.32 75.72
## 78 75.86 75.74 75.61 76.33 75.99 76.28 76.33 76.47 76.40 76.75
## 79 76.87 77.00 77.34 76.92 77.45 77.18 77.03 77.38 77.23 77.59
## 80 78.24 78.00 78.23 77.84 78.46 78.48 78.13 78.28 78.61 78.62
## 81 79.29 78.88 78.75 79.19 79.54 79.38 79.06 79.24 79.16 79.45
## 82 79.56 80.05 79.75 80.34 80.38 80.04 80.14 80.45 80.45 80.07
## 83 81.08 81.21 81.05 81.21 81.21 81.10 81.24 81.54 81.46 81.68
## 84 82.37 82.27 82.19 81.59 82.25 82.31 82.09 82.06 82.58 82.06
## 85 82.95 82.50 82.93 82.96 83.33 82.94 82.97 83.21 83.00 83.72
## 86 83.37 84.00 84.17 84.19 83.89 84.25 84.36 84.55 84.43 84.48
## 87 84.45 84.88 84.75 85.10 84.94 84.92 85.22 85.36 85.69 85.20
## 88 85.74 85.77 85.76 85.66 86.20 86.14 85.96 86.59 86.69 86.31
## 89 86.77 86.80 86.41 86.06 86.50 86.82 86.96 87.10 87.43 87.10
## 90 87.18 87.67 87.64 87.87 87.99 88.05 88.01 87.81 87.88 87.96
## 91 88.91 89.04 88.48 88.67 88.92 88.98 89.02 89.47 88.86 89.24
## 92 89.63 89.88 90.09 89.86 90.12 90.03 90.19 89.82 90.03 90.18
## 93 90.27 90.58 91.03 90.30 90.98 91.07 91.36 91.00 91.25 91.28
## 94 91.83 91.70 91.65 91.59 91.85 91.71 91.83 92.14 91.94 92.08
## 95 92.52 91.90 92.47 92.37 92.93 92.94 93.05 93.32 92.91 93.71
## 96 93.68 94.02 93.46 94.15 93.87 93.38 94.32 93.83 93.83 94.17
## 97 94.66 94.28 94.70 95.08 95.30 95.30 95.04 94.91 95.05 94.66
## 98 95.50 95.49 95.71 95.08 95.57 95.84 95.71 95.86 96.11 96.14
## 99 96.24 95.94 96.85 97.11 96.57 96.73 96.64 97.08 97.54 97.06
## 100 96.97 97.23 97.53 97.49 97.31 97.74 98.04 98.11 97.92 98.31
## 101 98.11 98.77 98.15 98.86 98.36 98.22 99.01 99.05 98.69 99.01
## 102 99.69 98.90 99.74 99.12 99.68 99.58 99.54 99.40 99.78 100.18
## 103 99.91 100.53 100.34 100.55 101.27 100.71 101.02 100.95 100.19 100.84
## 104 101.12 101.86 101.48 101.13 101.44 101.96 101.70 101.92 102.04 102.16
## 105 102.02 102.15 101.77 102.54 102.52 102.36 102.59 102.05 102.70 102.96
## 106 103.50 103.31 103.76 103.75 103.66 103.33 103.65 103.82 103.85 103.93
## 107 104.16 104.48 103.98 104.03 104.86 104.38 104.64 104.93 104.82 104.99
## 108 104.73 104.28 105.54 105.56 105.51 105.33 105.86 105.68 105.56 105.94
## 109 106.02 106.68 106.21 105.92 106.15 106.39 106.30 106.30 106.30 106.74
## 110 107.33 107.28 107.50 106.80 107.40 107.63 107.66 107.53 107.57 107.85
## 111 107.62 107.95 108.27 108.08 108.78 108.57 108.17 108.83 108.74 109.22
## 112 108.53 108.44 109.46 109.13 109.56 108.83 109.41 109.11 109.40 109.46
## 113 109.88 109.79 110.98 110.52 110.20 110.63 110.94 111.20 110.34 110.65
## 114 111.05 111.14 110.55 110.93 110.99 111.50 111.28 111.31 111.73 111.27
## 115 112.13 112.20 111.95 112.34 112.42 112.56 112.45 112.44 112.66 112.54
## 116 112.15 113.43 113.17 112.80 113.39 113.49 113.11 113.19 113.66 113.63
## 117 114.03 114.24 114.04 113.95 114.70 114.63 114.80 114.37 113.82 114.70
## 118 114.90 115.31 115.18 114.68 115.43 115.25 115.63 115.32 115.73 115.90
## 119 115.65 115.78 115.86 115.59 116.14 115.79 116.00 116.52 116.75 116.16
## 120 116.73 117.11 116.59 116.54 117.09 116.58 117.45 117.82 117.31 117.55
## 121 118.18 117.34 117.79 117.48 117.82 118.43 118.05 117.82 118.33 118.49
## 122 118.34 119.07 119.06 119.55 118.69 119.68 119.31 119.81 119.37 119.87
## 123 119.36 119.63 119.96 120.28 119.57 119.82 120.44 119.78 120.14 120.13
## 124 120.30 120.82 120.17 120.60 121.36 120.83 121.33 121.32 121.28 121.38
## 125 121.82 121.13 121.85 121.89 122.56 122.24 122.09 121.84 122.02 122.62
## 126 122.51 122.21 122.68 122.96 122.89 122.75 123.09 123.12 123.06 123.52
## 127 122.63 123.18 123.14 124.08 123.73 124.46 124.52 123.64 124.39 124.41
## 128 124.44 125.28 125.38 124.74 125.40 124.95 125.19 125.39 125.15 125.73
## 129 125.42 125.70 125.06 125.54 125.20 125.11 125.93 126.28 126.28 126.70
## 130 126.97 126.50 126.99 127.28 126.88 126.95 126.87 127.06 126.72 127.39
## 131 127.12 127.48 127.23 127.91 128.00 127.81 127.78 128.42 128.07 128.27
## 132 127.49 128.25 128.62 129.06 129.32 129.06 128.98 128.75 129.20 129.23
## 133 128.96 129.89 129.96 129.59 129.57 130.37 129.86 130.78 130.31 129.90
## 134 129.69 130.50 130.60 130.92 130.80 130.20 130.91 130.88 131.35 131.33
## 135 131.78 130.81 131.38 132.20 131.38 131.92 131.86 131.83 131.82 132.52
## 136 131.56 132.64 131.98 132.23 132.65 132.65 132.73 132.60 133.50 133.28
## 137 133.46 133.50 134.09 133.57 133.85 133.03 134.36 133.86 134.09 133.90
## 138 134.34 134.28 133.98 134.28 135.09 134.96 134.98 134.43 135.05 134.51
## 139 135.67 134.89 135.25 135.18 135.87 136.01 135.36 135.99 135.06 136.40
## 140 135.52 135.88 136.03 136.32 136.23 136.65 137.05 136.83 136.74 137.39
## 141 136.99 137.85 137.18 137.45 137.55 137.62 137.82 137.70 138.04 137.87
## 142 137.92 138.39 138.69 138.28 138.46 138.31 138.92 138.76 138.46 139.22
## 143 138.32 139.12 139.91 138.68 139.04 139.40 139.18 139.75 139.60 139.39
## 144 139.62 139.98 139.84 139.89 139.47 141.41 140.80 140.87 140.36 140.74
## 145 141.31 141.15 141.46 141.15 141.95 141.51 142.14 141.72 141.28 141.53
## 146 142.49 141.61 141.97 141.55 143.05 142.23 142.48 142.67 143.34 143.16
## 147 142.67 143.04 142.79 143.39 143.54 143.56 143.59 143.39 143.76 143.79
## 148 143.94 143.74 144.09 144.03 144.03 145.28 144.78 145.16 144.36 144.97
## 149 144.57 145.10 145.72 145.44 144.78 145.63 145.70 146.23 145.67 145.42
## 150 144.09 145.26 146.31 146.29 146.14 146.65 147.22 146.31 146.90 147.50
## 151 147.67 147.27 147.35 147.04 146.35 147.51 147.79 147.73 147.74 148.51
## 152 147.89 148.48 148.36 148.13 147.85 148.64 148.46 148.42 148.75 148.58
## 153 148.33 148.68 149.24 148.85 149.03 149.22 149.74 149.94 149.83 149.85
## 154 148.72 149.42 150.33 150.29 150.30 150.45 150.47 150.06 150.87 150.21
## 155 150.02 150.27 150.60 151.57 151.02 151.77 152.28 152.16 151.75 151.73
## 156 151.70 151.62 151.78 151.70 152.32 152.20 152.98 152.63 152.20 152.45
## 157 151.70 152.19 152.33 153.43 153.26 153.01 153.29 153.98 154.08 153.38
## 158 153.79 154.46 154.25 154.07 153.23 154.24 155.31 154.50 154.49 154.60
## 159 154.73 154.10 154.58 153.74 155.11 155.20 155.23 154.59 155.23 155.74
## 160 155.70 155.59 155.59 155.45 156.40 156.68 156.34 156.56 156.85 156.86
## 41 42 43 44 45 46 47 48 49 50
## 1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
## 2 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
## 3 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
## 4 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00
## 5 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
## 6 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00
## 7 6.98 7.00 7.00 7.00 7.00 7.00 7.00 7.00 6.99 7.00
## 8 7.98 7.99 7.99 8.00 8.00 8.00 8.00 7.99 8.00 8.00
## 9 9.00 9.00 8.98 9.00 8.99 9.00 9.00 9.00 8.97 8.99
## 10 9.95 9.98 10.00 9.99 9.99 10.00 9.97 9.96 10.00 9.99
## 11 10.99 10.99 10.98 11.00 11.00 10.98 10.99 11.00 10.98 10.99
## 12 11.97 11.92 11.96 11.99 11.99 11.98 11.98 11.99 11.99 12.00
## 13 12.91 12.94 12.95 12.97 12.94 12.97 12.95 12.98 12.97 12.94
## 14 13.98 13.95 13.96 13.97 13.96 13.98 13.98 13.93 13.95 13.96
## 15 14.92 14.91 14.97 14.94 14.94 14.95 14.97 14.90 14.95 14.94
## 16 15.90 15.92 15.94 15.90 15.91 15.94 15.92 15.88 15.95 15.92
## 17 16.90 16.91 16.91 16.86 16.95 16.93 16.95 16.93 16.97 16.96
## 18 17.87 17.93 17.92 17.87 17.93 17.93 17.95 17.87 17.93 17.94
## 19 18.87 18.93 18.85 18.87 18.86 18.98 18.95 18.95 18.87 18.91
## 20 19.86 19.84 19.87 19.81 19.90 19.89 19.94 19.91 19.88 19.87
## 21 20.86 20.88 20.85 20.87 20.96 20.86 20.93 20.90 20.90 20.86
## 22 21.72 21.89 21.85 21.86 21.80 21.84 21.87 21.94 21.85 21.84
## 23 22.73 22.91 22.82 22.73 22.87 22.86 22.87 22.90 22.91 22.90
## 24 23.76 23.72 23.80 23.85 23.89 23.82 23.81 23.73 23.87 23.85
## 25 24.75 24.82 24.74 24.78 24.80 24.82 24.82 24.84 24.81 24.88
## 26 25.62 25.75 25.82 25.82 25.75 25.71 25.87 25.80 25.75 25.78
## 27 26.68 26.73 26.79 26.77 26.82 26.81 26.80 26.72 26.87 26.76
## 28 27.64 27.64 27.74 27.73 27.76 27.76 27.73 27.79 27.76 27.77
## 29 28.63 28.70 28.71 28.73 28.73 28.71 28.73 28.80 28.79 28.83
## 30 29.68 29.56 29.64 29.63 29.66 29.80 29.68 29.71 29.78 29.81
## 31 30.54 30.58 30.66 30.77 30.67 30.71 30.66 30.69 30.72 30.74
## 32 31.66 31.77 31.61 31.61 31.65 31.67 31.64 31.72 31.74 31.71
## 33 32.51 32.59 32.59 32.68 32.67 32.72 32.63 32.75 32.67 32.82
## 34 33.44 33.63 33.65 33.71 33.72 33.62 33.64 33.73 33.65 33.69
## 35 34.52 34.54 34.54 34.74 34.61 34.56 34.62 34.69 34.71 34.61
## 36 35.61 35.67 35.58 35.57 35.59 35.45 35.56 35.70 35.72 35.75
## 37 36.42 36.57 36.57 36.51 36.53 36.61 36.50 36.65 36.64 36.52
## 38 37.37 37.46 37.57 37.52 37.68 37.61 37.64 37.56 37.49 37.78
## 39 38.62 38.42 38.43 38.71 38.73 38.49 38.62 38.54 38.61 38.68
## 40 39.34 39.58 39.61 39.49 39.58 39.68 39.56 39.50 39.48 39.60
## 41 40.22 40.40 40.56 40.54 40.50 40.51 40.67 40.56 40.52 40.53
## 42 41.28 41.48 41.39 41.49 41.50 41.56 41.39 41.59 41.58 41.64
## 43 42.40 42.45 42.59 42.45 42.65 42.47 42.43 42.34 42.51 42.59
## 44 43.21 43.48 43.54 43.46 43.47 43.47 43.43 43.35 43.67 43.52
## 45 44.36 44.54 44.43 44.34 44.46 44.47 44.53 44.40 44.57 44.68
## 46 45.21 45.45 45.37 45.46 45.54 45.59 45.48 45.37 45.44 45.47
## 47 46.28 46.20 46.27 46.27 46.44 46.62 46.39 46.57 46.48 46.62
## 48 47.28 47.36 47.40 47.30 47.40 47.53 47.53 47.66 47.38 47.58
## 49 48.27 48.23 48.19 48.23 48.45 48.51 48.36 48.42 48.37 48.40
## 50 49.19 49.39 49.13 49.41 49.36 49.15 49.44 49.40 49.34 49.32
## 51 50.25 50.24 50.38 50.41 50.12 50.20 50.40 50.44 50.34 50.29
## 52 51.13 51.23 51.41 51.29 51.43 51.34 51.24 51.38 51.26 51.28
## 53 52.24 52.35 52.24 52.20 52.21 52.27 52.33 52.33 52.41 52.44
## 54 53.27 53.24 53.40 53.27 53.21 53.22 53.10 53.21 53.25 53.43
## 55 53.84 54.16 54.29 54.35 54.27 54.23 54.21 54.37 54.40 54.23
## 56 55.09 55.29 55.09 55.09 55.02 55.22 55.09 55.49 55.21 55.38
## 57 56.23 56.19 56.13 56.12 56.15 56.19 56.12 56.35 56.19 56.32
## 58 56.97 57.45 57.19 57.26 57.21 57.29 57.43 57.25 57.43 57.37
## 59 57.89 58.22 57.98 58.01 58.01 58.32 58.11 58.24 58.23 58.34
## 60 59.00 59.04 58.96 59.25 59.27 59.15 59.13 58.87 59.16 59.19
## 61 60.09 60.00 60.21 60.10 60.06 59.97 60.18 59.88 60.16 60.37
## 62 60.96 60.74 61.00 60.99 61.18 61.22 61.17 61.03 61.27 61.07
## 63 62.06 61.98 62.05 62.35 62.07 62.16 61.95 62.09 62.15 62.22
## 64 62.79 62.83 62.90 62.98 62.91 63.12 62.84 63.03 63.26 63.63
## 65 63.81 64.08 63.90 64.15 64.09 64.07 63.78 64.01 63.98 64.01
## 66 65.20 64.85 64.94 64.90 64.92 65.13 65.07 65.15 65.06 65.14
## 67 65.81 65.93 65.73 65.79 65.97 66.04 65.74 65.95 66.02 66.25
## 68 67.11 66.83 66.91 67.06 66.95 66.79 67.06 67.09 66.84 66.93
## 69 67.72 67.92 67.54 67.88 67.99 67.98 67.80 68.16 67.81 67.94
## 70 68.50 68.92 68.61 69.00 68.80 68.88 69.04 68.91 68.77 69.13
## 71 70.14 69.91 69.69 69.59 70.05 70.02 70.08 70.05 70.14 70.04
## 72 70.78 70.81 70.65 70.96 70.87 70.80 70.68 70.62 70.92 71.06
## 73 71.86 72.00 71.47 71.71 71.87 71.91 72.15 72.18 71.89 72.09
## 74 72.96 72.79 72.77 73.05 72.87 72.61 72.78 72.97 73.28 72.85
## 75 73.61 73.93 73.71 73.79 73.76 73.70 73.56 73.91 73.77 73.82
## 76 74.42 74.46 74.81 74.32 74.87 75.12 75.01 74.94 74.65 74.81
## 77 75.58 75.38 75.87 75.83 75.79 75.83 75.58 75.73 75.84 75.92
## 78 76.31 76.78 76.54 76.57 76.80 76.81 76.74 76.49 76.70 76.95
## 79 77.51 77.33 77.62 77.66 77.96 77.50 77.76 77.74 77.74 77.59
## 80 78.70 78.66 78.77 78.52 78.82 78.77 78.83 78.87 78.93 78.95
## 81 79.18 79.68 79.31 79.85 79.74 79.86 79.98 79.71 79.45 80.05
## 82 80.40 80.46 80.67 80.69 80.50 80.84 81.08 80.66 80.87 80.65
## 83 81.66 81.64 81.70 81.62 81.77 81.70 81.55 81.74 81.81 82.05
## 84 82.00 82.45 82.50 82.56 82.76 82.58 82.54 82.59 82.79 82.90
## 85 83.25 83.63 83.23 83.62 83.61 83.33 83.59 83.72 83.75 83.85
## 86 84.22 84.48 84.57 84.03 84.65 84.29 84.41 84.86 84.38 84.79
## 87 85.37 85.36 85.19 85.39 85.92 85.78 85.94 85.26 85.61 85.83
## 88 86.13 86.22 86.43 86.64 86.63 86.57 86.37 86.69 86.93 86.65
## 89 87.42 87.25 87.39 87.41 87.66 87.45 87.58 87.63 87.76 87.65
## 90 88.59 88.43 88.43 88.53 88.33 87.93 88.18 88.63 88.60 88.69
## 91 89.66 89.24 89.37 89.28 89.45 89.58 89.95 89.54 89.53 89.87
## 92 89.75 90.62 90.44 90.47 90.21 90.59 90.53 90.75 90.37 90.55
## 93 91.31 91.47 91.26 91.43 91.56 91.62 91.42 91.84 91.67 91.52
## 94 92.31 92.07 92.49 92.18 92.51 92.43 92.50 92.52 92.60 92.52
## 95 93.08 93.65 93.03 93.40 93.21 93.67 93.33 93.90 93.12 93.77
## 96 94.37 94.22 94.04 93.89 94.60 94.93 94.33 94.68 94.27 94.66
## 97 95.44 95.13 94.61 95.25 95.61 95.57 95.28 95.40 95.63 95.13
## 98 96.02 96.45 95.82 96.42 96.13 96.20 96.48 96.65 96.41 96.44
## 99 97.63 97.29 96.98 97.32 97.52 97.38 97.17 97.61 97.57 97.38
## 100 98.02 98.35 98.29 98.32 98.54 98.40 98.69 98.57 98.72 98.28
## 101 99.25 98.89 98.91 99.48 99.14 99.11 99.19 99.66 99.21 99.60
## 102 99.65 100.10 100.46 100.26 100.13 100.38 100.29 99.96 100.29 100.15
## 103 100.82 101.05 100.85 101.53 100.75 101.23 101.68 101.35 101.52 101.35
## 104 102.28 102.39 101.59 102.20 101.95 102.09 102.15 102.64 102.33 102.49
## 105 103.14 103.05 102.98 102.84 102.99 102.94 103.02 103.46 103.21 102.98
## 106 103.45 104.13 103.91 104.25 104.23 103.97 104.41 104.18 104.61 104.54
## 107 104.98 104.71 105.11 105.24 105.14 104.97 105.18 105.11 105.30 105.65
## 108 105.76 105.89 106.19 106.31 106.02 105.82 105.95 106.27 106.26 106.58
## 109 106.86 106.86 107.02 107.22 106.73 107.08 106.76 107.15 107.37 107.12
## 110 107.97 108.30 107.79 107.53 108.01 108.41 108.06 107.67 108.39 108.57
## 111 109.03 108.73 109.04 109.04 108.85 108.95 109.36 108.97 109.01 109.54
## 112 109.58 109.31 109.52 110.11 110.20 110.42 109.97 110.30 110.39 110.06
## 113 111.11 110.78 110.80 110.89 111.37 111.29 110.78 111.07 111.33 111.16
## 114 111.65 111.77 111.82 111.78 111.81 111.42 112.17 112.23 112.04 112.39
## 115 112.38 112.80 112.68 112.90 113.31 112.83 112.93 113.07 113.30 113.11
## 116 114.02 113.90 113.59 113.78 114.14 114.23 114.35 113.76 113.92 114.02
## 117 114.96 114.55 114.84 115.32 114.59 114.81 115.18 115.16 115.23 114.82
## 118 115.10 115.71 115.67 116.04 116.33 116.07 116.30 116.09 116.22 116.33
## 119 116.75 116.67 116.78 116.95 116.66 116.96 116.73 116.83 116.95 117.29
## 120 117.62 117.60 117.83 117.95 117.56 117.79 117.94 118.09 117.96 118.29
## 121 118.52 118.95 118.40 118.71 118.92 118.65 119.00 119.17 119.02 119.15
## 122 119.59 119.56 119.26 120.18 119.74 120.06 120.05 119.71 120.28 119.88
## 123 120.45 120.19 120.63 120.75 120.45 120.77 120.61 121.16 120.81 120.68
## 124 121.93 121.78 121.86 121.64 121.68 121.92 121.28 122.21 121.98 121.98
## 125 122.40 122.83 122.64 122.41 122.70 122.53 123.15 122.85 122.76 122.79
## 126 123.51 123.48 123.61 122.89 123.77 123.96 123.64 123.83 123.80 124.08
## 127 124.52 124.64 124.33 124.78 124.93 124.85 124.70 124.89 124.90 125.04
## 128 125.49 125.57 125.58 125.59 125.81 125.65 126.06 125.82 125.44 125.61
## 129 126.67 126.12 126.50 127.04 126.93 126.49 126.83 126.38 126.75 126.99
## 130 127.10 127.16 127.06 127.24 127.61 127.88 127.36 127.80 127.95 127.80
## 131 128.31 128.80 128.49 128.50 129.06 128.60 128.75 128.77 128.57 128.97
## 132 129.39 129.10 128.73 129.25 129.45 129.41 129.38 130.13 129.97 130.06
## 133 130.58 130.17 130.57 130.80 130.91 130.88 130.33 130.78 131.13 131.13
## 134 131.04 131.06 131.17 131.54 131.51 131.47 131.69 131.32 132.09 131.79
## 135 132.58 132.65 132.05 132.19 132.26 132.89 132.95 133.12 132.78 132.70
## 136 133.35 133.25 133.24 133.29 133.45 133.50 133.62 133.76 133.95 134.29
## 137 134.28 134.53 133.74 133.82 134.37 134.67 134.86 135.01 134.45 135.18
## 138 135.04 135.18 135.59 135.15 135.55 135.59 135.39 136.04 135.84 135.71
## 139 136.27 135.96 136.29 135.97 136.44 136.28 136.93 136.19 136.78 136.87
## 140 136.98 136.80 137.36 137.28 137.26 137.23 137.27 137.82 137.16 137.89
## 141 138.22 137.46 137.96 138.32 138.68 138.85 139.05 138.67 139.05 138.62
## 142 138.44 139.66 139.67 139.90 139.27 139.24 139.56 138.81 139.81 139.35
## 143 140.57 140.44 140.21 140.09 140.44 140.56 140.51 140.47 140.14 140.89
## 144 141.11 141.31 141.19 141.07 141.57 142.11 141.86 141.30 141.55 141.58
## 145 141.59 141.88 142.19 142.26 142.39 142.58 142.24 141.84 142.70 142.71
## 146 142.53 143.52 142.84 144.11 143.22 143.33 143.25 143.45 142.94 143.40
## 147 144.36 143.93 143.83 143.69 144.83 143.84 144.49 144.32 144.79 144.94
## 148 144.74 144.67 145.36 144.70 145.53 145.22 145.67 144.96 145.49 145.25
## 149 146.25 146.00 146.24 146.59 145.69 146.47 146.33 146.05 146.17 146.16
## 150 146.92 146.91 147.08 147.28 147.29 147.70 147.56 147.81 147.45 147.20
## 151 148.00 148.38 148.11 148.45 148.38 148.41 148.26 148.49 148.45 148.65
## 152 148.83 149.23 148.99 149.94 149.32 149.30 149.76 149.88 149.28 149.13
## 153 148.64 149.75 150.10 150.08 150.02 150.04 149.87 150.47 150.35 149.89
## 154 150.63 151.12 151.15 150.81 150.89 151.13 151.27 151.53 151.06 151.76
## 155 152.00 151.80 151.96 151.85 152.61 152.07 151.59 152.21 152.67 152.52
## 156 153.14 153.02 153.33 153.11 153.30 152.72 153.38 153.55 153.34 153.24
## 157 153.73 153.88 153.76 153.47 154.02 153.97 154.19 154.39 154.69 154.27
## 158 154.27 154.73 154.19 154.98 155.14 154.98 154.78 155.28 155.51 155.38
## 159 156.13 156.08 155.71 155.97 155.63 155.72 156.31 156.39 156.11 156.04
## 160 156.81 157.16 156.90 156.79 157.04 156.56 157.30 157.09 157.04 157.56
## 51 52 53 54 55 56 57 58 59 60
## 1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
## 2 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
## 3 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
## 4 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00
## 5 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
## 6 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00
## 7 7.00 7.00 7.00 7.00 7.00 7.00 7.00 7.00 7.00 7.00
## 8 8.00 8.00 7.98 8.00 8.00 8.00 8.00 8.00 8.00 8.00
## 9 8.97 8.99 9.00 9.00 8.99 9.00 9.00 8.99 9.00 9.00
## 10 9.99 10.00 10.00 10.00 10.00 10.00 10.00 10.00 9.98 10.00
## 11 10.98 11.00 10.99 11.00 10.99 10.98 11.00 10.98 10.99 11.00
## 12 12.00 11.99 11.99 12.00 11.99 11.98 12.00 11.99 11.98 12.00
## 13 12.99 12.99 12.99 12.99 12.94 12.97 12.98 12.99 13.00 12.98
## 14 13.98 13.94 14.00 13.94 14.00 13.98 13.99 13.99 13.97 13.99
## 15 14.97 14.99 14.98 14.99 14.97 14.97 14.98 14.98 14.97 15.00
## 16 15.99 15.96 16.00 15.96 15.96 15.96 15.98 15.96 15.98 15.97
## 17 16.95 16.88 16.92 16.97 16.96 16.94 16.97 16.99 16.96 16.95
## 18 17.97 17.94 17.96 17.89 17.94 17.95 17.97 17.97 17.97 17.98
## 19 18.92 18.87 18.96 18.91 18.97 18.93 18.95 18.97 18.98 18.95
## 20 19.83 19.94 19.96 19.91 19.93 19.96 19.98 19.94 19.96 19.98
## 21 20.94 20.91 20.89 20.94 20.96 20.91 20.97 20.94 20.94 20.93
## 22 21.94 21.84 21.85 21.85 21.93 21.91 21.89 21.92 21.97 21.92
## 23 22.90 22.91 22.85 22.85 22.89 22.93 22.84 22.89 22.92 22.91
## 24 23.78 23.86 23.88 23.88 23.93 23.85 23.86 23.92 23.89 23.92
## 25 24.86 24.76 24.90 24.87 24.87 24.87 24.85 24.88 24.85 24.95
## 26 25.83 25.88 25.78 25.85 25.90 25.84 25.94 25.87 25.85 25.90
## 27 26.88 26.76 26.92 26.83 26.81 26.85 26.94 26.82 26.90 26.92
## 28 27.85 27.77 27.79 27.76 27.88 27.79 27.85 27.85 27.85 27.83
## 29 28.79 28.80 28.78 28.78 28.87 28.87 28.90 28.86 28.83 28.87
## 30 29.72 29.70 29.80 29.88 29.80 29.82 29.81 29.85 29.82 29.78
## 31 30.71 30.82 30.67 30.76 30.75 30.87 30.79 30.79 30.91 30.82
## 32 31.67 31.65 31.77 31.75 31.76 31.78 31.80 31.79 31.82 31.80
## 33 32.73 32.74 32.74 32.71 32.80 32.78 32.82 32.79 32.79 32.74
## 34 33.73 33.68 33.68 33.60 33.79 33.86 33.74 33.76 33.84 33.80
## 35 34.79 34.65 34.76 34.76 34.80 34.66 34.84 34.72 34.77 34.80
## 36 35.76 35.61 35.60 35.69 35.84 35.64 35.60 35.74 35.76 35.78
## 37 36.70 36.76 36.66 36.66 36.69 36.70 36.65 36.84 36.61 36.77
## 38 37.73 37.57 37.75 37.65 37.71 37.69 37.66 37.81 37.74 37.81
## 39 38.59 38.60 38.67 38.78 38.68 38.72 38.80 38.61 38.65 38.78
## 40 39.72 39.62 39.61 39.64 39.78 39.68 39.74 39.60 39.66 39.77
## 41 40.74 40.57 40.68 40.73 40.62 40.76 40.65 40.74 40.73 40.76
## 42 41.53 41.59 41.59 41.48 41.68 41.63 41.67 41.75 41.78 41.77
## 43 42.52 42.63 42.69 42.61 42.58 42.70 42.74 42.61 42.66 42.56
## 44 43.70 43.56 43.52 43.53 43.58 43.77 43.65 43.66 43.60 43.49
## 45 44.56 44.59 44.57 44.57 44.46 44.53 44.56 44.54 44.68 44.54
## 46 45.51 45.39 45.58 45.58 45.69 45.56 45.50 45.69 45.56 45.59
## 47 46.62 46.47 46.49 46.53 46.58 46.68 46.60 46.58 46.58 46.62
## 48 47.64 47.52 47.50 47.34 47.48 47.61 47.49 47.54 47.45 47.68
## 49 48.34 48.47 48.42 48.59 48.47 48.51 48.66 48.42 48.50 48.69
## 50 49.29 49.40 49.43 49.47 49.34 49.63 49.54 49.62 49.41 49.58
## 51 50.39 50.46 50.41 50.30 50.67 50.35 50.45 50.41 50.55 50.42
## 52 51.47 51.37 51.41 51.32 51.35 51.53 51.38 51.50 51.40 51.53
## 53 52.31 52.50 52.59 52.44 52.53 52.64 52.45 52.52 52.49 52.46
## 54 53.28 53.43 53.31 53.40 53.50 53.44 53.56 53.45 53.56 53.52
## 55 54.33 54.33 54.45 54.34 54.48 54.36 54.54 54.35 54.43 54.52
## 56 55.24 55.29 55.43 55.36 55.48 55.51 55.38 55.48 55.51 55.44
## 57 56.44 56.28 56.32 56.45 56.26 56.44 56.36 56.54 56.52 56.54
## 58 57.20 57.18 57.44 57.30 57.32 57.26 57.40 57.34 57.49 57.49
## 59 58.30 58.30 58.38 58.46 58.32 58.43 58.36 58.55 58.36 58.50
## 60 59.26 59.32 59.21 59.38 59.26 59.39 59.47 59.51 59.61 59.54
## 61 60.29 60.32 60.17 60.46 60.34 60.31 60.32 60.34 60.27 60.39
## 62 61.34 61.12 61.41 61.30 61.23 61.36 61.30 61.31 61.30 61.48
## 63 62.11 62.05 62.14 62.20 62.32 62.55 62.28 62.42 62.44 62.34
## 64 63.33 63.06 63.36 63.18 63.43 63.18 63.41 63.41 63.30 63.35
## 65 64.17 64.49 64.28 64.27 64.31 64.28 64.17 64.56 64.31 64.30
## 66 64.73 65.25 65.14 65.12 65.23 65.13 65.27 65.10 65.43 65.40
## 67 66.09 66.10 65.93 66.40 66.20 66.32 66.14 66.39 66.36 66.35
## 68 67.20 67.20 67.01 67.22 67.28 67.36 67.30 67.27 67.19 67.20
## 69 68.30 68.25 68.12 68.31 68.08 68.18 68.27 68.20 68.25 68.40
## 70 69.01 69.19 69.17 69.26 69.07 69.25 69.23 69.11 69.23 69.29
## 71 70.19 70.16 70.07 70.22 70.04 70.14 70.19 70.01 70.18 70.27
## 72 71.16 71.23 70.83 71.10 71.21 71.23 71.34 71.14 71.13 71.30
## 73 71.96 72.05 72.17 71.91 72.20 72.05 72.15 72.22 72.12 72.15
## 74 73.00 73.21 72.81 73.04 73.25 73.02 73.14 73.21 73.06 73.14
## 75 73.84 73.91 73.88 74.00 74.12 73.96 74.12 74.07 73.99 74.36
## 76 74.93 75.10 75.12 75.01 75.10 74.86 75.14 75.25 75.24 75.02
## 77 76.19 76.00 75.98 76.10 76.11 76.28 76.09 76.23 76.27 75.88
## 78 76.98 77.11 77.06 77.05 76.82 76.91 77.08 76.91 77.18 77.19
## 79 77.87 77.90 78.31 77.94 77.89 77.93 78.21 78.23 78.12 78.34
## 80 78.69 79.10 79.16 78.92 79.02 78.93 79.03 79.17 79.17 79.18
## 81 79.56 79.93 79.72 80.30 80.05 80.08 79.93 80.14 80.24 80.29
## 82 80.65 80.86 80.74 80.66 80.80 80.89 81.33 81.01 81.21 81.23
## 83 81.86 81.85 81.73 82.10 81.85 81.85 81.98 81.98 82.15 82.13
## 84 82.40 82.80 83.01 82.40 82.94 82.97 82.83 82.99 82.82 83.18
## 85 83.87 83.66 84.01 83.76 84.18 84.05 84.25 83.85 84.08 84.00
## 86 84.83 84.62 85.10 84.74 84.67 85.03 84.90 84.79 85.02 85.28
## 87 85.93 85.99 85.56 85.94 86.02 85.71 85.85 86.07 86.06 86.09
## 88 86.83 86.78 86.84 86.98 86.75 86.80 86.87 86.50 86.89 86.92
## 89 88.12 87.83 87.92 87.88 87.85 87.94 87.59 87.58 87.93 88.16
## 90 88.71 88.87 88.86 88.90 88.81 89.04 88.89 88.58 88.76 88.63
## 91 89.96 89.49 89.81 89.79 89.70 89.72 89.63 89.66 89.95 90.04
## 92 90.80 90.96 90.92 90.71 90.70 90.97 90.86 90.75 90.86 91.04
## 93 91.59 91.75 91.59 91.71 91.96 91.81 91.93 92.01 91.95 91.92
## 94 92.60 92.83 92.98 92.68 92.68 92.84 92.88 92.45 92.41 92.86
## 95 93.64 93.44 93.62 93.49 93.66 94.01 93.79 93.97 93.80 93.83
## 96 94.48 94.30 94.61 94.85 94.73 94.61 94.78 94.96 94.48 94.83
## 97 95.61 95.57 95.82 95.68 95.54 95.54 95.38 95.89 95.55 95.69
## 98 96.54 96.76 96.65 96.74 96.73 97.00 96.74 96.74 96.77 96.88
## 99 97.59 97.45 97.54 97.75 97.40 97.69 97.66 97.59 97.58 97.97
## 100 98.61 98.42 98.63 98.46 98.71 98.81 98.72 98.51 98.94 98.72
## 101 99.09 99.29 99.67 99.62 99.37 99.66 99.87 99.89 99.64 99.69
## 102 100.61 100.70 100.36 100.93 100.87 100.57 100.91 100.64 100.78 100.95
## 103 101.72 101.44 101.18 101.59 101.71 101.63 101.27 101.55 101.82 101.86
## 104 102.66 102.62 102.71 102.68 102.78 102.21 102.39 102.88 102.83 102.60
## 105 103.33 103.53 104.12 103.74 103.77 103.73 103.52 103.64 104.06 103.67
## 106 104.32 104.68 104.63 104.41 104.65 104.50 104.24 104.82 104.75 104.84
## 107 105.48 105.23 105.37 105.66 105.71 105.58 105.51 105.53 105.80 105.82
## 108 106.68 106.38 106.65 106.02 106.35 106.34 106.39 106.80 106.63 106.73
## 109 107.46 107.30 107.13 107.47 107.23 107.87 107.37 107.43 107.80 107.42
## 110 108.14 108.31 108.46 108.01 108.55 108.77 108.22 108.64 108.66 108.43
## 111 109.21 109.09 109.58 109.68 109.55 109.24 109.62 109.47 109.69 109.45
## 112 110.39 109.91 110.62 110.45 110.30 110.48 110.55 110.51 110.69 110.53
## 113 111.15 110.84 111.16 111.51 111.09 111.62 111.64 111.61 111.49 111.49
## 114 112.16 112.42 112.39 112.47 112.38 112.50 112.58 112.59 112.68 112.83
## 115 113.41 113.53 113.60 113.33 113.36 113.44 113.82 113.34 113.61 113.77
## 116 114.67 114.19 114.13 114.41 114.41 114.58 114.47 114.45 114.53 114.40
## 117 115.12 115.24 115.03 115.52 115.61 115.47 115.34 115.46 115.19 115.79
## 118 116.31 115.81 115.96 116.00 116.09 116.30 116.48 116.54 116.29 116.17
## 119 117.02 117.27 117.21 117.26 117.66 116.98 117.60 116.96 117.37 117.52
## 120 118.13 118.21 118.07 118.11 118.52 118.51 118.27 118.46 118.51 118.28
## 121 119.32 119.18 119.25 119.47 119.15 119.30 119.07 119.39 119.43 119.18
## 122 120.06 120.41 120.38 120.37 120.20 120.47 120.47 120.21 119.87 120.38
## 123 120.61 120.94 120.99 121.26 121.70 121.36 121.23 121.38 121.53 121.66
## 124 121.70 122.10 121.95 122.24 122.45 121.96 122.33 122.37 122.29 122.19
## 125 123.24 123.12 123.07 123.21 123.23 123.07 123.39 123.24 123.62 123.40
## 126 124.11 123.93 124.27 124.41 124.18 124.50 124.42 124.36 124.22 124.44
## 127 125.21 124.81 125.54 125.22 124.62 125.07 125.57 125.12 125.24 125.34
## 128 125.87 126.22 125.48 125.96 126.04 126.28 126.17 126.58 126.15 126.43
## 129 126.97 126.92 127.02 126.72 127.47 127.23 127.36 127.40 127.28 127.19
## 130 128.34 127.57 128.29 128.56 127.99 128.27 128.50 128.48 128.16 127.90
## 131 128.45 128.97 129.17 129.25 129.46 129.54 129.47 129.17 129.44 129.06
## 132 129.76 129.67 130.35 130.12 130.27 130.32 129.98 130.46 130.46 129.93
## 133 130.97 131.01 130.86 131.41 130.82 131.09 131.36 131.23 131.69 130.89
## 134 131.82 132.01 132.01 132.24 131.91 132.30 132.64 132.03 132.34 132.00
## 135 132.88 132.90 132.49 133.35 132.99 133.30 132.88 133.32 133.13 133.51
## 136 133.95 134.20 133.76 134.27 133.63 133.87 133.60 133.89 134.13 134.06
## 137 134.92 135.07 135.13 134.88 135.04 135.04 134.91 135.39 135.36 135.31
## 138 135.76 136.20 135.95 136.17 136.19 136.42 136.10 136.03 136.56 136.47
## 139 136.77 136.35 136.89 136.92 136.73 137.14 136.99 136.87 137.29 137.49
## 140 137.51 137.42 137.94 138.03 138.14 137.82 138.14 137.91 138.30 137.92
## 141 138.59 138.51 138.23 139.45 138.64 138.94 138.96 139.13 139.28 139.17
## 142 139.92 140.00 140.17 140.00 139.83 140.12 140.17 140.13 140.19 140.22
## 143 140.63 140.86 140.46 141.11 140.61 141.09 141.15 140.60 140.94 141.03
## 144 142.15 141.59 142.23 142.04 141.72 141.43 142.22 142.10 142.45 142.08
## 145 142.80 142.61 143.02 142.81 142.61 143.11 143.04 142.76 143.02 143.16
## 146 143.60 143.82 144.01 143.94 143.80 143.87 143.95 143.78 144.26 144.09
## 147 144.90 144.74 145.12 144.86 144.83 145.37 144.79 145.07 144.95 145.06
## 148 145.66 145.63 145.57 145.46 145.61 145.78 146.16 146.12 146.35 146.18
## 149 146.33 147.17 146.94 146.76 147.19 147.22 146.77 146.86 147.19 146.66
## 150 147.67 147.19 147.75 147.84 147.84 147.38 147.84 147.61 148.13 147.65
## 151 148.88 148.43 148.57 148.97 149.10 148.59 149.21 148.87 148.87 149.15
## 152 149.31 149.38 149.62 149.94 150.04 149.89 149.01 150.21 150.01 149.85
## 153 150.57 150.69 150.37 150.61 150.50 150.91 150.87 150.43 150.85 150.71
## 154 151.45 151.76 151.77 151.61 151.56 151.66 151.85 151.44 151.60 152.17
## 155 152.49 152.78 152.19 152.83 152.75 152.95 152.16 153.20 152.89 153.19
## 156 153.67 153.65 153.61 153.33 154.27 154.02 153.86 153.94 153.80 154.57
## 157 154.12 154.51 154.55 154.60 154.69 154.68 155.07 154.54 155.06 154.45
## 158 155.04 155.14 155.22 155.95 155.45 155.45 155.45 155.98 156.03 155.93
## 159 156.16 157.03 156.76 156.30 156.53 156.65 156.68 156.23 157.19 157.01
## 160 157.71 158.04 157.46 157.64 157.40 157.75 157.91 157.23 157.49 157.77
## 61 62 63 64 65 66 67 68 69 70
## 1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
## 2 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
## 3 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
## 4 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00
## 5 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
## 6 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00
## 7 7.00 7.00 6.99 7.00 7.00 7.00 7.00 7.00 7.00 7.00
## 8 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00
## 9 9.00 9.00 9.00 9.00 8.99 9.00 9.00 9.00 9.00 9.00
## 10 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 9.99
## 11 11.00 10.99 11.00 11.00 11.00 10.99 11.00 11.00 11.00 11.00
## 12 12.00 11.99 12.00 12.00 11.99 12.00 12.00 12.00 12.00 12.00
## 13 12.98 12.99 13.00 12.99 12.99 13.00 12.99 13.00 12.98 13.00
## 14 13.99 13.98 13.99 13.99 13.98 13.98 13.96 14.00 13.98 13.99
## 15 14.99 14.96 15.00 14.98 14.99 14.97 15.00 14.98 14.99 15.00
## 16 15.98 15.94 15.98 15.98 16.00 15.98 15.99 15.99 15.96 16.00
## 17 16.95 16.99 16.97 16.96 16.99 16.98 16.98 16.99 16.96 17.00
## 18 17.94 17.96 17.95 18.00 17.96 17.98 17.99 17.97 17.97 17.99
## 19 18.95 18.95 18.97 18.98 19.00 18.95 18.96 18.96 18.99 18.97
## 20 19.97 19.99 19.98 19.98 19.97 19.98 19.99 19.97 19.96 19.96
## 21 20.93 20.95 20.95 20.94 20.92 20.94 20.95 21.00 20.95 20.95
## 22 21.92 21.92 21.91 21.94 21.96 21.91 21.98 21.97 21.95 21.93
## 23 22.91 22.91 22.86 22.92 22.99 22.91 22.96 22.95 22.93 22.99
## 24 23.88 23.86 23.92 23.96 23.92 23.91 23.94 23.95 23.96 23.96
## 25 24.90 24.86 24.87 24.90 24.94 24.92 24.93 24.94 24.93 24.98
## 26 25.91 25.89 25.87 25.89 25.89 25.89 25.93 25.88 25.95 25.92
## 27 26.87 26.89 26.93 26.88 26.85 26.89 26.91 26.91 26.88 26.88
## 28 27.84 27.86 27.88 27.89 27.90 27.87 27.87 27.94 27.90 27.87
## 29 28.87 28.89 28.90 28.87 28.84 28.87 28.91 28.94 28.84 28.89
## 30 29.87 29.89 29.75 29.87 29.85 29.91 29.88 29.90 29.85 29.93
## 31 30.91 30.89 30.85 30.93 30.86 30.87 30.83 30.87 30.91 30.86
## 32 31.82 31.79 31.91 31.86 31.86 31.84 31.83 31.79 31.89 31.91
## 33 32.79 32.77 32.75 32.78 32.87 32.86 32.81 32.86 32.80 32.85
## 34 33.78 33.79 33.83 33.81 33.79 33.85 33.80 33.83 33.81 33.83
## 35 34.81 34.72 34.85 34.76 34.82 34.80 34.79 34.84 34.87 34.87
## 36 35.77 35.80 35.77 35.75 35.86 35.91 35.77 35.84 35.83 35.94
## 37 36.77 36.79 36.77 36.76 36.82 36.85 36.79 36.76 36.86 36.83
## 38 37.64 37.77 37.84 37.84 37.78 37.82 37.81 37.71 37.89 37.73
## 39 38.69 38.70 38.77 38.76 38.72 38.68 38.88 38.76 38.75 38.77
## 40 39.65 39.78 39.70 39.72 39.80 39.76 39.89 39.75 39.80 39.77
## 41 40.64 40.77 40.68 40.74 40.76 40.77 40.77 40.74 40.74 40.68
## 42 41.74 41.68 41.79 41.66 41.66 41.76 41.71 41.79 41.75 41.71
## 43 42.59 42.72 42.72 42.78 42.69 42.69 42.67 42.77 42.71 42.75
## 44 43.70 43.66 43.62 43.63 43.74 43.71 43.73 43.67 43.82 43.77
## 45 44.69 44.65 44.59 44.63 44.70 44.69 44.73 44.68 44.75 44.74
## 46 45.57 45.64 45.56 45.72 45.63 45.72 45.68 45.75 45.65 45.71
## 47 46.66 46.60 46.56 46.61 46.63 46.57 46.78 46.73 46.77 46.79
## 48 47.58 47.80 47.71 47.60 47.68 47.80 47.65 47.72 47.65 47.68
## 49 48.62 48.70 48.59 48.58 48.61 48.53 48.70 48.64 48.68 48.66
## 50 49.64 49.58 49.67 49.66 49.62 49.44 49.70 49.69 49.69 49.83
## 51 50.45 50.67 50.70 50.54 50.61 50.50 50.54 50.65 50.69 50.75
## 52 51.59 51.70 51.55 51.57 51.62 51.55 51.64 51.69 51.72 51.61
## 53 52.67 52.51 52.50 52.47 52.61 52.49 52.70 52.58 52.66 52.64
## 54 53.52 53.52 53.54 53.51 53.63 53.69 53.55 53.48 53.69 53.57
## 55 54.48 54.57 54.63 54.59 54.63 54.60 54.60 54.62 54.71 54.57
## 56 55.59 55.50 55.56 55.68 55.57 55.42 55.54 55.65 55.66 55.49
## 57 56.55 56.64 56.45 56.44 56.43 56.58 56.62 56.66 56.54 56.59
## 58 57.54 57.55 57.50 57.54 57.54 57.50 57.55 57.52 57.52 57.60
## 59 58.58 58.52 58.50 58.20 58.54 58.48 58.53 58.48 58.42 58.64
## 60 59.37 59.48 59.45 59.52 59.58 59.57 59.57 59.54 59.57 59.72
## 61 60.51 60.59 60.37 60.51 60.36 60.52 60.52 60.65 60.51 60.43
## 62 61.46 61.34 61.39 61.39 61.47 61.59 61.47 61.49 61.50 61.51
## 63 62.12 62.25 62.46 62.46 62.40 62.26 62.47 62.47 62.62 62.62
## 64 63.40 63.33 63.48 63.38 63.50 63.33 63.41 63.29 63.42 63.64
## 65 64.49 64.36 64.38 64.47 64.35 64.50 64.36 64.50 64.42 64.47
## 66 65.49 65.36 65.49 65.48 65.29 65.35 65.44 65.44 65.49 65.38
## 67 66.28 66.37 66.49 66.21 66.41 66.53 66.27 66.32 66.43 66.68
## 68 67.39 67.40 67.16 67.43 67.53 67.28 67.38 67.37 67.35 67.44
## 69 68.32 68.36 68.14 68.42 68.27 68.27 68.36 68.41 68.56 68.37
## 70 69.39 69.35 69.32 69.58 69.18 69.40 69.28 69.32 69.28 69.30
## 71 70.45 70.21 70.42 70.30 70.24 70.37 70.26 70.30 70.28 70.50
## 72 71.20 71.19 71.17 71.39 71.28 71.28 71.10 71.26 71.37 71.34
## 73 72.05 72.36 72.42 72.33 72.39 72.32 72.12 72.51 72.33 72.43
## 74 73.19 72.98 73.33 73.28 73.21 73.29 73.25 73.40 73.33 73.34
## 75 74.19 74.21 74.16 74.30 74.15 74.29 74.54 74.33 74.42 74.17
## 76 75.13 75.21 75.23 75.29 75.35 75.16 75.26 75.46 75.29 75.30
## 77 76.18 76.24 76.49 76.22 76.25 76.29 76.21 76.07 76.32 76.01
## 78 77.20 77.03 77.12 77.32 77.16 77.27 77.27 77.25 77.37 77.27
## 79 78.22 78.14 78.06 78.08 78.40 78.21 78.33 78.28 78.31 78.25
## 80 79.32 79.08 79.19 79.27 79.14 79.04 79.34 79.39 79.28 79.35
## 81 80.10 80.18 80.29 80.24 80.15 80.45 80.21 80.22 80.23 80.32
## 82 80.92 81.33 81.09 81.29 81.44 81.15 81.24 81.19 81.18 81.18
## 83 82.07 81.96 82.07 81.88 82.04 82.30 82.34 82.28 81.95 82.17
## 84 82.88 82.95 83.24 83.33 83.05 83.27 83.33 83.17 83.23 83.44
## 85 83.97 84.18 83.95 84.02 84.10 84.22 84.06 84.13 84.17 84.31
## 86 85.24 85.05 84.97 85.18 85.06 85.10 85.01 85.00 85.30 85.08
## 87 86.01 85.89 85.96 86.21 86.25 86.11 86.13 86.23 86.35 86.20
## 88 87.27 86.96 86.98 87.16 86.96 86.89 87.00 87.30 87.21 87.17
## 89 88.27 87.63 88.14 87.91 88.15 88.10 88.08 88.28 88.25 88.14
## 90 88.98 88.92 88.88 88.97 89.05 89.12 89.29 89.29 89.21 89.38
## 91 89.85 90.11 89.96 89.91 90.01 90.11 90.08 89.99 89.85 90.23
## 92 90.74 90.83 90.80 91.03 90.93 90.85 91.24 91.13 91.05 91.24
## 93 92.03 92.09 92.01 91.91 91.86 92.07 92.16 91.94 92.28 92.03
## 94 93.19 92.63 93.02 92.85 93.01 93.32 93.06 92.95 93.24 93.13
## 95 94.01 93.79 94.21 94.09 94.08 93.95 94.17 93.88 94.08 94.17
## 96 94.95 94.63 94.92 95.08 94.91 95.13 95.13 95.05 95.13 95.12
## 97 95.81 96.03 96.05 95.92 95.67 95.84 95.87 96.11 96.21 96.17
## 98 97.14 96.60 96.88 96.67 96.85 97.16 97.38 96.94 96.92 97.00
## 99 98.00 97.91 97.94 97.97 97.80 97.93 98.00 98.10 97.81 98.17
## 100 99.00 98.79 99.11 99.09 99.00 98.86 98.85 98.89 99.16 99.19
## 101 99.92 99.75 99.81 100.00 100.11 99.89 100.16 99.93 100.04 100.13
## 102 100.47 100.90 100.71 100.56 101.00 100.93 100.55 101.11 101.10 100.79
## 103 101.97 101.76 101.67 101.68 101.76 101.72 101.79 101.96 101.93 102.05
## 104 102.85 102.81 102.91 102.79 102.55 102.86 103.17 102.90 103.26 103.19
## 105 104.10 103.68 103.85 103.88 103.75 103.93 104.00 103.58 103.75 104.05
## 106 104.64 104.70 104.99 104.95 104.80 104.79 104.70 104.84 105.07 104.92
## 107 105.69 105.88 105.66 105.88 105.97 105.98 105.70 106.15 105.82 106.04
## 108 106.62 106.48 106.95 106.85 106.80 106.91 106.71 106.95 106.89 107.13
## 109 107.50 107.66 107.87 107.67 107.96 108.07 107.86 107.61 107.95 107.89
## 110 108.86 108.53 108.77 108.78 108.74 108.78 108.86 108.79 108.91 109.02
## 111 109.79 109.48 109.66 109.66 109.79 109.93 109.79 109.82 109.71 109.97
## 112 110.68 110.72 110.58 110.89 110.76 110.83 110.72 110.80 110.72 111.08
## 113 111.61 111.69 111.75 111.72 111.64 111.95 111.68 111.97 111.69 111.85
## 114 112.55 112.42 112.88 112.30 112.81 113.02 112.69 113.02 112.95 112.84
## 115 113.45 113.88 113.70 113.69 113.48 113.70 113.66 113.69 113.72 114.04
## 116 114.69 114.62 114.84 114.64 114.73 114.69 114.68 114.65 114.77 114.95
## 117 115.76 115.61 115.43 115.40 115.72 115.60 115.64 115.73 115.78 115.99
## 118 116.76 116.84 116.62 116.24 116.59 117.02 116.77 117.07 116.70 116.85
## 119 117.22 117.29 117.73 117.28 117.69 117.72 117.71 117.78 117.82 117.41
## 120 118.77 118.75 118.40 118.43 118.82 118.46 118.66 118.54 118.77 118.69
## 121 119.37 119.52 119.12 119.40 119.63 119.74 119.85 119.58 119.67 119.76
## 122 119.96 120.68 120.52 120.78 120.58 120.69 120.38 120.78 120.32 120.67
## 123 121.56 121.47 121.58 121.72 121.64 121.62 121.58 121.80 121.69 121.51
## 124 122.50 122.50 122.37 122.93 122.44 122.64 122.68 122.71 122.82 122.49
## 125 123.10 123.43 123.57 123.56 123.56 123.82 123.76 123.45 123.72 123.46
## 126 124.46 124.43 124.67 124.47 124.45 124.48 124.49 124.54 124.75 124.79
## 127 125.40 125.14 125.40 125.83 125.44 125.83 125.43 125.51 125.65 125.90
## 128 126.70 125.92 126.55 126.51 126.65 126.18 126.71 126.70 126.43 126.73
## 129 127.49 127.47 127.62 127.74 127.13 127.76 127.47 127.66 127.41 127.70
## 130 128.44 128.47 128.45 128.73 128.24 128.14 128.30 128.47 128.80 128.68
## 131 129.27 129.62 129.55 129.30 129.60 129.36 129.53 129.30 129.48 129.54
## 132 130.16 130.41 130.30 130.47 130.74 130.17 130.72 130.37 130.81 130.56
## 133 131.23 131.22 131.45 131.62 131.27 131.54 131.16 131.38 132.07 131.55
## 134 132.29 132.72 132.33 132.31 132.20 132.61 132.74 132.51 132.23 133.02
## 135 133.16 133.31 133.30 133.20 133.18 133.53 133.09 133.62 133.68 133.70
## 136 134.11 134.04 134.74 134.47 134.41 134.43 134.57 134.42 134.84 134.52
## 137 135.49 135.50 135.24 135.48 135.35 135.58 135.25 135.69 135.74 135.58
## 138 136.20 136.37 136.26 136.51 136.37 136.62 136.57 136.43 136.73 136.37
## 139 137.15 137.31 137.53 137.53 137.52 137.16 137.36 137.31 137.81 137.46
## 140 138.19 138.29 138.15 138.17 138.23 138.30 138.35 138.23 138.21 138.29
## 141 139.32 139.40 139.22 139.20 139.16 139.58 139.58 139.15 139.17 139.40
## 142 140.22 140.49 140.37 140.45 140.18 140.34 140.39 140.35 140.63 140.44
## 143 140.85 141.15 141.32 141.00 141.35 141.53 141.47 141.57 141.39 141.75
## 144 142.27 142.48 142.11 142.20 142.28 142.49 142.53 142.22 142.55 142.61
## 145 142.51 143.16 143.11 143.30 143.32 143.05 143.28 143.59 143.64 143.35
## 146 144.30 143.90 143.89 144.19 144.14 144.08 144.37 144.19 144.36 143.78
## 147 145.13 144.94 145.09 145.38 145.17 145.10 145.30 145.49 145.14 145.77
## 148 146.45 145.87 146.32 146.40 146.05 146.17 146.21 146.55 146.28 146.51
## 149 147.08 147.09 147.09 147.13 147.22 147.38 147.41 147.29 147.09 147.49
## 150 148.40 148.23 147.97 148.24 148.34 148.49 148.20 148.15 148.13 148.43
## 151 149.28 149.28 148.95 148.85 149.48 149.10 149.03 149.01 148.54 149.57
## 152 149.37 150.02 149.84 149.85 150.38 150.18 150.27 150.28 150.39 150.22
## 153 151.05 151.08 151.20 151.09 151.15 151.12 151.44 151.57 151.55 151.29
## 154 151.72 151.66 152.20 152.23 152.29 152.41 152.01 152.05 152.24 152.55
## 155 153.04 152.52 152.73 153.11 153.07 152.94 153.42 152.85 152.82 153.30
## 156 154.09 153.70 153.76 154.29 153.60 154.42 153.97 154.08 153.81 153.69
## 157 155.31 154.84 155.15 154.86 155.17 155.25 155.19 155.25 155.11 155.06
## 158 156.53 156.23 156.06 155.39 156.16 155.93 156.05 156.21 156.22 156.06
## 159 156.88 156.97 156.85 157.14 156.83 157.51 156.96 157.38 157.42 157.36
## 160 157.94 158.00 158.03 158.16 158.42 157.99 158.14 158.17 158.45 158.24
## 71 72 73 74 75 76 77 78 79 80
## 1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
## 2 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
## 3 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
## 4 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00
## 5 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
## 6 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00
## 7 7.00 7.00 7.00 6.99 7.00 7.00 7.00 7.00 7.00 7.00
## 8 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00
## 9 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00
## 10 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00
## 11 11.00 11.00 11.00 11.00 11.00 11.00 11.00 11.00 11.00 10.99
## 12 12.00 12.00 12.00 12.00 12.00 11.99 11.99 12.00 12.00 12.00
## 13 12.99 12.99 13.00 13.00 13.00 13.00 13.00 12.99 13.00 12.99
## 14 13.99 14.00 13.99 14.00 13.98 14.00 14.00 13.99 14.00 14.00
## 15 14.99 15.00 14.99 15.00 15.00 15.00 14.96 14.99 14.99 15.00
## 16 15.97 16.00 15.98 15.98 15.98 15.99 15.99 16.00 15.99 16.00
## 17 16.96 17.00 16.98 17.00 16.98 16.99 16.99 16.98 16.99 16.97
## 18 17.99 17.99 17.97 17.98 17.97 17.98 17.99 18.00 17.97 18.00
## 19 18.95 18.98 18.97 18.94 18.97 18.98 18.95 18.96 18.96 18.99
## 20 19.98 19.99 19.95 19.97 19.98 19.97 19.94 20.00 19.98 19.98
## 21 20.97 20.98 20.97 20.96 20.96 20.99 20.98 20.96 20.96 21.00
## 22 21.93 21.95 21.94 21.95 21.99 21.96 21.95 21.98 21.96 21.99
## 23 22.96 22.97 22.96 22.95 22.96 22.98 22.99 22.95 22.99 22.98
## 24 23.96 23.94 23.95 23.94 23.94 23.91 23.96 23.95 23.95 23.98
## 25 24.90 24.92 24.92 24.96 24.92 24.94 24.96 24.98 24.97 24.94
## 26 25.93 25.93 25.91 25.92 25.94 25.91 25.93 25.92 25.92 25.94
## 27 26.92 26.89 26.90 26.92 26.97 26.94 26.90 26.95 26.91 26.91
## 28 27.95 27.94 27.92 27.91 27.93 27.91 27.93 27.87 27.94 27.98
## 29 28.92 28.91 28.92 28.93 28.92 28.94 28.91 28.92 28.95 28.92
## 30 29.95 29.90 29.86 29.88 29.92 29.90 29.90 29.90 29.93 29.90
## 31 30.89 30.93 30.90 30.96 30.92 30.84 30.92 30.89 30.92 30.86
## 32 31.93 31.91 31.90 31.94 31.90 31.86 31.91 31.88 31.86 31.94
## 33 32.85 32.87 32.84 32.89 32.86 32.93 32.92 32.88 32.86 32.87
## 34 33.83 33.88 33.86 33.87 33.86 33.91 33.88 33.89 33.89 33.88
## 35 34.91 34.92 34.87 34.93 34.85 34.89 34.95 34.87 34.84 34.89
## 36 35.81 35.78 35.84 35.78 35.91 35.89 35.85 35.87 35.97 35.92
## 37 36.82 36.82 36.86 36.84 36.88 36.80 36.87 36.87 36.82 36.90
## 38 37.82 37.81 37.87 37.72 37.86 37.86 37.86 37.85 37.89 37.89
## 39 38.82 38.85 38.81 38.86 38.79 38.76 38.88 38.88 38.86 38.91
## 40 39.81 39.74 39.84 39.85 39.88 39.86 39.88 39.74 39.84 39.84
## 41 40.85 40.83 40.74 40.72 40.73 40.81 40.80 40.82 40.78 40.83
## 42 41.82 41.81 41.71 41.79 41.80 41.85 41.78 41.80 41.86 41.81
## 43 42.77 42.77 42.77 42.75 42.84 42.76 42.68 42.84 42.74 42.72
## 44 43.84 43.76 43.70 43.76 43.75 43.88 43.73 43.81 43.89 43.86
## 45 44.79 44.81 44.86 44.81 44.72 44.71 44.77 44.85 44.77 44.79
## 46 45.72 45.81 45.76 45.76 45.82 45.82 45.80 45.81 45.77 45.80
## 47 46.66 46.78 46.77 46.73 46.73 46.69 46.84 46.77 46.74 46.80
## 48 47.75 47.72 47.67 47.59 47.67 47.80 47.74 47.75 47.67 47.82
## 49 48.74 48.75 48.75 48.70 48.71 48.70 48.65 48.74 48.83 48.76
## 50 49.77 49.67 49.71 49.75 49.63 49.64 49.84 49.78 49.74 49.70
## 51 50.56 50.65 50.62 50.82 50.64 50.61 50.63 50.75 50.81 50.78
## 52 51.68 51.63 51.71 51.68 51.57 51.68 51.60 51.72 51.72 51.76
## 53 52.60 52.57 52.54 52.79 52.76 52.73 52.64 52.78 52.76 52.76
## 54 53.65 53.74 53.60 53.73 53.64 53.67 53.56 53.71 53.64 53.64
## 55 54.62 54.57 54.59 54.65 54.61 54.62 54.71 54.66 54.80 54.70
## 56 55.61 55.51 55.60 55.56 55.61 55.76 55.61 55.71 55.68 55.70
## 57 56.53 56.66 56.61 56.60 56.68 56.72 56.59 56.70 56.70 56.67
## 58 57.57 57.62 57.60 57.63 57.54 57.59 57.61 57.66 57.70 57.56
## 59 58.57 58.54 58.61 58.63 58.62 58.61 58.56 58.68 58.60 58.63
## 60 59.60 59.58 59.59 59.69 59.71 59.66 59.75 59.61 59.69 59.61
## 61 60.48 60.50 60.59 60.70 60.66 60.64 60.64 60.65 60.76 60.56
## 62 61.37 61.61 61.60 61.69 61.57 61.63 61.72 61.69 61.66 61.61
## 63 62.69 62.58 62.61 62.61 62.54 62.60 62.61 62.72 62.66 62.56
## 64 63.47 63.62 63.50 63.43 63.51 63.65 63.62 63.66 63.69 63.65
## 65 64.34 64.43 64.47 64.67 64.49 64.54 64.65 64.60 64.67 64.52
## 66 65.33 65.59 65.48 65.50 65.52 65.45 65.39 65.53 65.61 65.73
## 67 66.53 66.31 66.51 66.59 66.48 66.36 66.54 66.46 66.51 66.65
## 68 67.52 67.53 67.49 67.44 67.66 67.61 67.59 67.54 67.66 67.55
## 69 68.38 68.52 68.55 68.36 68.54 68.60 68.49 68.53 68.47 68.61
## 70 69.38 69.47 69.49 69.49 69.41 69.62 69.47 69.54 69.61 69.50
## 71 70.40 70.40 70.36 70.52 70.56 70.31 70.35 70.54 70.40 70.42
## 72 71.43 71.44 71.44 71.55 71.54 71.45 71.39 71.47 71.48 71.43
## 73 72.41 72.36 72.44 72.47 72.33 72.46 72.40 72.61 72.57 72.44
## 74 73.54 73.27 73.56 73.59 73.46 73.38 73.45 73.42 73.47 73.52
## 75 74.38 74.25 74.37 74.33 74.42 74.29 74.45 74.39 74.60 74.37
## 76 75.55 75.43 75.34 75.39 75.57 75.45 75.42 75.58 75.43 75.44
## 77 76.23 76.28 76.42 76.25 76.42 76.50 76.45 76.27 76.42 76.41
## 78 77.59 77.20 77.33 77.41 77.52 77.35 77.41 77.37 77.48 77.37
## 79 78.22 78.35 78.44 78.49 78.44 78.45 78.37 78.43 78.32 78.54
## 80 79.33 79.44 79.13 79.47 79.20 79.28 79.20 79.50 79.38 79.42
## 81 80.25 80.55 80.36 80.40 80.21 80.39 80.10 80.38 80.33 80.36
## 82 81.25 81.01 81.20 81.20 81.34 81.30 81.35 81.49 81.28 81.32
## 83 82.16 82.23 82.17 82.42 82.30 82.36 82.18 82.47 82.18 82.33
## 84 83.40 83.15 83.09 83.30 83.49 83.17 83.32 83.45 83.37 83.26
## 85 84.17 84.34 84.18 84.16 84.20 84.23 84.41 84.53 84.38 84.33
## 86 85.43 85.15 85.26 85.38 85.25 85.43 85.07 85.31 85.50 85.25
## 87 86.32 86.07 86.27 86.09 86.45 86.32 86.27 86.39 86.27 86.24
## 88 87.40 87.29 87.01 87.36 87.36 87.36 87.07 87.20 87.38 87.36
## 89 88.15 88.34 88.09 88.31 88.25 88.39 88.32 88.25 88.39 88.26
## 90 89.24 89.27 89.32 89.17 89.03 89.25 89.30 89.34 89.11 89.21
## 91 90.01 90.26 90.18 90.20 90.25 90.28 90.10 90.45 90.34 90.29
## 92 91.36 91.14 91.04 91.21 91.20 91.14 91.33 91.43 91.41 91.36
## 93 92.19 92.23 92.05 92.19 92.23 92.14 92.45 92.23 92.40 92.36
## 94 93.05 93.27 93.20 93.08 93.30 93.09 93.28 93.34 93.28 93.03
## 95 94.15 94.20 94.27 94.21 94.21 94.18 94.04 94.15 94.28 94.29
## 96 95.13 95.11 95.07 94.90 95.02 95.25 95.32 95.03 95.23 95.03
## 97 95.98 96.08 96.01 96.15 95.84 96.03 96.24 96.07 96.03 96.16
## 98 96.94 97.27 97.26 97.04 97.12 97.11 96.99 97.32 97.08 97.31
## 99 98.23 98.21 97.92 98.24 97.96 98.01 98.15 98.36 97.87 98.18
## 100 98.84 99.09 99.02 99.02 99.02 99.13 99.16 98.95 99.09 99.22
## 101 99.98 100.13 100.04 99.90 100.08 100.05 100.22 100.03 100.01 100.37
## 102 101.13 100.80 101.10 100.90 100.91 101.05 101.16 101.19 101.33 101.12
## 103 101.80 102.07 101.88 102.21 102.08 102.27 102.09 102.28 102.09 102.17
## 104 103.17 103.13 103.13 103.14 102.88 103.05 103.04 103.08 102.98 103.02
## 105 104.03 103.81 104.19 104.02 104.27 104.14 104.16 103.83 104.08 104.23
## 106 104.90 105.01 104.86 104.96 105.19 105.07 105.06 105.02 105.43 105.01
## 107 106.08 105.99 106.22 105.97 106.15 106.10 105.75 106.15 106.29 106.23
## 108 106.57 106.95 107.09 107.11 107.09 107.18 107.21 107.08 107.22 107.05
## 109 107.95 108.06 108.20 108.11 108.13 108.00 108.03 108.05 107.76 108.09
## 110 108.73 108.78 108.72 109.03 109.20 109.14 108.93 108.92 109.17 108.84
## 111 110.03 109.72 109.86 109.93 109.98 109.94 110.18 109.78 109.99 109.92
## 112 111.01 111.01 110.64 111.02 110.93 111.16 111.00 111.17 111.02 111.02
## 113 111.90 112.07 111.86 112.05 112.11 111.97 111.69 112.17 111.96 112.16
## 114 112.98 112.88 113.08 113.00 113.03 113.16 113.06 113.29 112.96 113.18
## 115 113.74 114.11 113.98 113.98 113.95 114.05 114.09 113.88 113.97 114.07
## 116 114.84 114.75 115.07 114.78 115.09 114.90 115.07 115.14 115.09 114.84
## 117 116.01 116.00 115.74 115.98 115.78 115.70 115.91 115.85 116.04 115.99
## 118 116.99 117.07 116.81 116.94 116.91 116.58 116.82 117.13 117.16 116.97
## 119 117.98 117.44 117.70 118.02 117.48 118.10 117.86 117.99 117.77 117.68
## 120 118.65 118.59 118.80 118.75 118.76 119.33 119.07 118.76 119.01 118.80
## 121 119.75 119.65 119.90 119.85 119.61 120.08 119.74 119.82 119.91 119.71
## 122 120.80 120.68 120.72 120.72 120.70 121.03 120.67 120.90 121.14 120.98
## 123 121.69 121.81 121.84 121.77 121.62 121.58 121.93 122.06 121.86 121.96
## 124 122.48 122.88 122.77 122.95 122.76 122.78 122.66 122.85 123.03 122.84
## 125 123.54 124.04 123.97 123.65 123.76 123.70 123.66 123.91 123.73 123.72
## 126 124.58 125.01 124.64 124.79 124.58 124.64 124.79 125.05 124.88 124.78
## 127 125.86 125.70 125.61 125.78 125.85 125.55 125.88 125.70 125.59 125.93
## 128 126.93 126.84 126.60 126.72 126.76 126.87 127.20 126.89 126.72 126.95
## 129 127.91 127.78 127.71 127.31 127.75 127.68 127.71 127.74 127.75 127.72
## 130 128.35 128.40 128.76 128.79 128.83 128.79 128.84 128.90 128.72 128.85
## 131 129.61 129.61 129.59 129.29 129.66 129.53 129.65 129.55 129.65 130.00
## 132 130.65 130.64 130.39 130.77 131.05 130.80 130.80 130.83 131.05 130.86
## 133 131.67 131.55 131.46 131.68 131.72 131.83 131.79 131.50 131.92 131.84
## 134 132.73 132.45 133.00 132.74 132.61 132.62 132.53 132.76 132.57 132.91
## 135 133.50 134.02 133.51 133.33 133.56 134.01 133.40 133.53 134.03 133.83
## 136 134.56 134.74 134.23 134.76 134.85 134.59 134.60 135.00 134.64 134.59
## 137 135.22 135.45 135.52 135.68 135.78 135.34 135.62 135.61 135.81 135.45
## 138 136.59 136.63 136.75 136.25 136.52 136.41 136.61 136.49 136.94 136.96
## 139 137.43 137.64 138.15 137.62 137.46 137.78 137.33 137.58 137.68 138.00
## 140 138.69 138.48 138.69 138.39 138.41 138.32 138.82 138.37 138.35 138.89
## 141 139.67 139.96 139.81 139.57 139.40 139.82 139.59 139.42 139.73 139.94
## 142 140.58 140.36 140.86 140.60 140.59 140.64 140.83 140.66 140.62 140.58
## 143 141.49 141.35 141.39 141.43 141.62 141.75 141.63 141.83 141.74 141.48
## 144 142.38 142.29 142.51 142.57 142.57 142.67 142.31 142.38 142.87 142.74
## 145 143.73 143.46 143.23 143.68 143.91 143.57 143.51 143.37 143.84 143.40
## 146 144.19 144.18 144.56 144.56 144.57 144.60 144.62 144.42 144.55 144.76
## 147 145.21 145.39 145.61 145.35 145.40 145.41 145.70 145.67 145.69 145.72
## 148 146.33 146.74 146.49 146.32 146.42 146.66 146.30 146.37 146.49 146.87
## 149 147.32 147.35 147.50 147.22 147.62 147.47 147.83 147.57 147.28 147.45
## 150 148.51 148.66 148.66 148.77 148.27 148.57 148.81 148.40 148.49 148.68
## 151 149.07 149.62 149.23 149.47 149.62 149.44 149.71 149.87 149.45 149.80
## 152 150.27 150.35 150.59 150.52 150.61 150.51 150.56 150.59 150.57 150.74
## 153 151.24 151.16 151.12 151.43 151.24 151.68 151.56 151.79 151.35 151.54
## 154 152.15 152.11 152.52 152.31 152.64 152.15 152.74 152.68 152.12 152.55
## 155 153.35 153.37 153.02 153.39 153.54 153.39 153.48 153.21 153.38 153.22
## 156 153.92 154.16 154.23 154.45 154.16 154.35 154.39 154.43 154.19 154.48
## 157 155.02 155.09 155.28 155.42 155.48 155.01 155.17 155.30 155.19 155.27
## 158 156.15 156.22 156.67 156.39 156.40 156.60 156.31 156.41 156.40 156.45
## 159 157.30 157.05 157.28 157.46 157.07 157.18 157.62 157.60 157.49 157.51
## 160 158.31 158.33 158.42 158.31 158.20 158.63 158.60 158.69 158.61 158.55
## 81 82 83 84 85 86 87 88 89 90
## 1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
## 2 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
## 3 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
## 4 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00
## 5 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
## 6 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00
## 7 7.00 7.00 7.00 7.00 7.00 7.00 7.00 7.00 7.00 7.00
## 8 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00
## 9 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00
## 10 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00
## 11 11.00 11.00 11.00 11.00 11.00 11.00 11.00 11.00 11.00 11.00
## 12 12.00 12.00 12.00 12.00 12.00 12.00 12.00 12.00 12.00 12.00
## 13 12.99 13.00 13.00 13.00 12.98 13.00 12.99 13.00 13.00 13.00
## 14 14.00 13.98 14.00 14.00 14.00 14.00 14.00 14.00 14.00 14.00
## 15 14.99 15.00 15.00 14.99 15.00 15.00 15.00 15.00 15.00 15.00
## 16 15.99 16.00 15.99 15.98 16.00 16.00 15.98 16.00 15.99 16.00
## 17 17.00 16.98 16.99 17.00 17.00 16.97 16.99 17.00 17.00 17.00
## 18 18.00 17.98 17.98 18.00 17.99 17.99 17.97 17.99 17.99 17.99
## 19 18.98 18.99 19.00 18.99 18.98 18.98 19.00 18.99 19.00 18.99
## 20 20.00 19.96 19.97 19.97 20.00 19.99 19.99 19.98 19.99 19.96
## 21 21.00 20.99 20.95 20.98 20.97 20.98 20.98 20.97 20.99 21.00
## 22 21.98 21.99 21.97 21.97 21.96 21.99 21.96 21.98 21.98 21.98
## 23 22.99 22.97 22.97 22.98 22.95 22.98 22.94 22.96 23.00 23.00
## 24 23.98 23.96 23.98 23.95 23.98 23.99 23.95 23.97 23.97 23.96
## 25 24.93 24.95 24.93 24.97 24.93 24.98 24.97 24.93 24.95 24.95
## 26 25.94 25.96 25.96 25.98 25.96 25.96 25.95 25.96 25.95 25.98
## 27 26.94 26.93 26.96 26.95 26.97 26.98 26.97 26.98 26.98 26.97
## 28 27.96 27.93 27.95 27.95 27.97 27.96 27.95 27.93 27.96 27.93
## 29 28.90 28.83 28.91 28.95 28.92 28.94 28.92 28.91 28.90 28.97
## 30 29.88 29.96 29.94 29.93 29.95 29.93 29.97 29.99 29.96 29.96
## 31 30.89 30.88 30.94 30.95 30.90 30.93 30.90 30.88 30.95 30.98
## 32 31.88 31.92 31.89 31.98 31.96 31.85 31.93 31.90 31.91 31.93
## 33 32.87 32.88 32.94 32.91 32.94 32.92 32.89 32.93 32.89 32.85
## 34 33.94 33.91 33.91 33.89 33.92 33.96 33.92 33.92 33.92 33.92
## 35 34.92 34.78 34.94 34.92 34.89 34.89 34.96 34.92 34.94 34.91
## 36 35.89 35.89 35.91 35.88 35.88 35.91 35.89 35.90 35.92 35.90
## 37 36.81 36.92 36.92 36.94 36.83 36.86 36.84 36.90 36.91 36.89
## 38 37.86 37.85 37.91 37.83 37.86 37.92 37.90 37.92 37.94 37.89
## 39 38.89 38.83 38.84 38.80 38.90 38.90 38.86 38.85 38.93 38.90
## 40 39.83 39.87 39.82 39.86 39.80 39.87 39.90 39.87 39.89 39.93
## 41 40.80 40.87 40.87 40.87 40.84 40.92 40.90 40.83 40.81 40.89
## 42 41.85 41.89 41.80 41.88 41.78 41.87 41.96 41.76 41.91 41.78
## 43 42.80 42.83 42.80 42.82 42.88 42.83 42.90 42.88 42.86 42.87
## 44 43.83 43.78 43.80 43.75 43.86 43.85 43.83 43.86 43.86 43.82
## 45 44.82 44.83 44.75 44.85 44.92 44.76 44.77 44.84 44.83 44.84
## 46 45.84 45.69 45.84 45.76 45.79 45.77 45.85 45.85 45.84 45.83
## 47 46.78 46.88 46.69 46.81 46.72 46.87 46.82 46.85 46.87 46.83
## 48 47.74 47.72 47.88 47.72 47.80 47.76 47.86 47.77 47.80 47.83
## 49 48.80 48.84 48.74 48.86 48.84 48.73 48.81 48.72 48.74 48.90
## 50 49.77 49.79 49.80 49.82 49.82 49.67 49.81 49.85 49.70 49.79
## 51 50.75 50.83 50.80 50.74 50.76 50.77 50.75 50.82 50.86 50.77
## 52 51.76 51.72 51.76 51.80 51.79 51.87 51.69 51.88 51.80 51.74
## 53 52.64 52.68 52.85 52.71 52.70 52.80 52.78 52.77 52.81 52.85
## 54 53.67 53.78 53.65 53.70 53.71 53.78 53.72 53.76 53.79 53.78
## 55 54.63 54.73 54.56 54.74 54.78 54.81 54.76 54.70 54.77 54.84
## 56 55.67 55.71 55.72 55.68 55.77 55.60 55.76 55.74 55.74 55.70
## 57 56.76 56.72 56.67 56.69 56.77 56.61 56.77 56.75 56.72 56.80
## 58 57.73 57.73 57.59 57.72 57.75 57.65 57.71 57.68 57.77 57.63
## 59 58.67 58.66 58.53 58.61 58.76 58.74 58.68 58.56 58.71 58.87
## 60 59.57 59.60 59.57 59.72 59.62 59.71 59.73 59.74 59.84 59.81
## 61 60.66 60.56 60.62 60.71 60.67 60.62 60.74 60.77 60.68 60.66
## 62 61.60 61.55 61.70 61.70 61.61 61.78 61.62 61.69 61.64 61.68
## 63 62.70 62.57 62.57 62.70 62.37 62.77 62.67 62.76 62.71 62.70
## 64 63.65 63.52 63.62 63.60 63.76 63.58 63.60 63.65 63.60 63.54
## 65 64.73 64.57 64.65 64.64 64.63 64.66 64.71 64.72 64.62 64.78
## 66 65.68 65.70 65.71 65.57 65.63 65.70 65.61 65.65 65.65 65.52
## 67 66.52 66.48 66.59 66.57 66.64 66.72 66.64 66.71 66.64 66.60
## 68 67.43 67.53 67.54 67.57 67.65 67.56 67.68 67.63 67.54 67.66
## 69 68.54 68.62 68.62 68.62 68.64 68.54 68.67 68.71 68.63 68.68
## 70 69.46 69.48 69.54 69.51 69.54 69.63 69.65 69.54 69.65 69.60
## 71 70.50 70.62 70.57 70.50 70.62 70.53 70.64 70.62 70.63 70.69
## 72 71.44 71.61 71.49 71.64 71.50 71.59 71.61 71.45 71.63 71.74
## 73 72.51 72.53 72.62 72.58 72.66 72.57 72.57 72.52 72.47 72.64
## 74 73.53 73.54 73.52 73.63 73.40 73.54 73.59 73.58 73.60 73.65
## 75 74.65 74.51 74.55 74.55 74.58 74.61 74.52 74.45 74.60 74.60
## 76 75.53 75.49 75.45 75.46 75.54 75.46 75.54 75.57 75.55 75.65
## 77 76.40 76.44 76.43 76.50 76.50 76.58 76.53 76.55 76.52 76.62
## 78 77.48 77.57 77.52 77.52 77.50 77.55 77.34 77.62 77.71 77.58
## 79 78.43 78.58 78.47 78.51 78.42 78.42 78.40 78.57 78.54 78.54
## 80 79.55 79.49 79.26 79.38 79.51 79.34 79.52 79.61 79.58 79.52
## 81 80.50 80.52 80.46 80.38 80.48 80.45 80.54 80.55 80.46 80.38
## 82 81.39 81.48 81.35 81.43 81.44 81.40 81.48 81.38 81.42 81.60
## 83 82.51 82.61 82.33 82.45 82.56 82.51 82.39 82.50 82.65 82.58
## 84 83.40 83.32 83.25 83.40 83.50 83.45 83.55 83.45 83.35 83.65
## 85 84.34 84.54 84.57 84.43 84.41 84.33 84.47 84.47 84.55 84.53
## 86 85.42 85.45 85.42 85.50 85.44 85.39 85.47 85.49 85.45 85.43
## 87 86.20 86.43 86.50 86.41 86.43 86.47 86.49 86.32 86.46 86.16
## 88 87.48 87.29 87.35 87.25 87.51 87.37 87.38 87.58 87.39 87.41
## 89 88.21 88.37 88.29 88.51 88.37 88.31 88.38 88.23 88.31 88.41
## 90 89.43 89.43 89.41 89.38 89.37 89.48 89.37 89.46 89.40 89.42
## 91 90.22 90.32 90.23 90.39 90.23 90.36 90.31 90.52 90.36 90.47
## 92 91.41 91.51 91.49 91.39 91.34 91.43 91.29 91.40 91.23 91.50
## 93 92.20 92.24 92.35 92.36 92.57 92.35 92.37 92.26 92.31 92.51
## 94 93.33 93.25 93.26 93.23 93.40 93.34 93.29 93.37 93.23 93.51
## 95 94.37 94.33 94.22 94.14 94.50 94.31 94.40 94.31 94.55 94.36
## 96 95.14 95.22 95.25 95.39 95.25 95.44 95.09 95.15 95.29 95.42
## 97 96.28 96.15 96.19 96.29 96.25 96.32 96.32 96.34 96.30 96.49
## 98 97.17 97.24 97.16 97.26 97.13 97.30 97.29 97.31 97.27 97.34
## 99 98.17 98.27 98.26 98.29 98.04 98.35 98.34 98.16 98.17 98.25
## 100 99.12 99.13 99.39 99.29 99.20 99.37 99.40 99.20 99.33 99.42
## 101 100.33 100.08 100.16 100.33 100.41 100.26 100.33 100.31 100.13 100.34
## 102 101.25 101.36 101.15 101.15 101.31 101.13 101.38 101.39 101.41 101.44
## 103 102.02 102.12 102.19 102.06 102.35 101.97 102.22 102.24 102.41 102.37
## 104 103.09 102.89 103.25 103.23 103.12 103.18 103.20 103.09 103.41 103.35
## 105 103.86 104.14 104.25 104.19 104.32 104.05 104.41 104.25 104.40 104.07
## 106 105.10 105.27 105.32 105.03 105.18 105.16 105.17 105.28 105.28 105.19
## 107 106.19 106.10 106.09 106.29 106.26 106.20 106.38 106.01 106.03 106.38
## 108 107.05 106.89 107.05 107.32 107.22 106.97 107.20 107.32 106.99 107.28
## 109 108.03 108.16 108.03 108.23 108.13 107.99 108.31 107.88 108.14 108.18
## 110 108.96 109.37 108.90 109.16 109.12 109.05 109.03 109.29 109.30 109.17
## 111 110.23 110.02 110.15 110.14 110.13 110.10 110.15 110.28 110.12 110.24
## 112 111.23 111.24 111.36 111.27 111.09 111.26 111.25 111.23 111.21 111.35
## 113 112.03 112.19 112.21 112.11 111.91 112.25 112.04 112.43 112.36 112.08
## 114 113.18 113.21 113.20 112.97 113.46 112.84 113.06 113.07 113.06 113.24
## 115 113.97 114.05 113.94 113.93 114.22 113.76 114.09 114.36 114.08 114.27
## 116 114.93 115.17 114.91 115.00 114.96 115.09 115.13 115.14 115.07 115.27
## 117 115.76 116.00 116.11 116.05 115.90 116.04 116.45 116.13 116.34 116.02
## 118 117.06 117.10 117.26 117.27 117.18 117.09 117.08 117.22 116.89 116.90
## 119 118.21 117.93 117.83 118.12 117.99 118.17 118.09 118.00 118.00 118.27
## 120 118.97 118.93 118.83 118.89 119.10 118.96 119.16 119.24 119.17 119.14
## 121 119.97 119.88 119.98 120.18 119.91 120.09 120.05 120.10 119.96 120.19
## 122 120.82 121.09 120.93 120.70 120.98 120.90 121.18 121.11 121.09 121.27
## 123 122.05 121.91 122.11 121.78 122.05 121.94 121.92 122.23 121.73 122.17
## 124 123.19 122.97 122.95 123.00 123.00 122.84 123.09 122.90 123.08 123.02
## 125 123.65 123.86 123.89 123.57 124.11 123.81 124.13 124.11 124.05 124.10
## 126 124.81 124.98 124.95 125.00 124.93 124.78 124.92 125.28 125.12 125.01
## 127 126.07 125.95 126.10 126.00 126.12 126.00 126.02 126.03 125.97 125.92
## 128 126.60 126.84 127.06 127.14 126.99 126.98 126.92 126.90 126.92 126.86
## 129 127.94 127.92 127.82 127.80 127.84 128.12 127.95 128.21 127.81 128.01
## 130 129.03 128.91 128.94 128.82 128.94 128.89 128.97 129.10 128.94 129.10
## 131 129.84 130.03 130.21 129.98 129.88 130.03 129.89 129.80 129.96 130.05
## 132 130.89 131.07 130.73 130.71 130.82 130.94 131.01 130.98 130.68 130.86
## 133 131.89 131.67 131.71 131.68 131.79 131.80 131.96 132.06 132.06 132.13
## 134 132.84 132.60 132.84 132.83 132.46 132.86 133.06 133.00 132.87 132.77
## 135 133.82 133.46 133.82 134.06 134.02 133.75 133.84 133.87 134.10 133.79
## 136 134.77 134.62 134.86 134.94 134.82 135.11 134.96 135.01 135.05 134.83
## 137 135.95 136.05 135.89 135.70 135.79 136.10 135.84 135.85 135.97 135.92
## 138 136.81 136.81 136.86 136.85 136.80 136.49 137.06 136.94 137.16 136.95
## 139 137.61 137.87 137.59 137.71 137.69 137.81 138.03 137.92 137.84 137.62
## 140 138.82 138.64 138.71 138.88 138.72 138.88 139.09 138.80 139.02 138.83
## 141 139.75 139.68 139.64 139.91 139.78 140.05 139.78 139.66 139.69 139.94
## 142 140.72 140.42 140.80 141.03 140.84 140.81 140.88 140.81 140.99 140.91
## 143 141.62 141.86 141.65 141.59 141.93 141.95 141.82 141.78 141.77 141.85
## 144 142.56 142.47 142.63 142.83 142.65 142.84 142.88 142.80 142.78 142.82
## 145 143.67 143.76 143.94 143.97 143.74 143.66 143.99 144.02 143.83 143.91
## 146 144.88 144.66 144.37 144.67 144.61 144.78 144.86 144.91 144.94 145.13
## 147 145.44 145.59 145.99 145.47 145.64 145.61 145.66 146.05 146.03 145.70
## 148 146.59 146.56 146.96 146.82 146.77 146.72 146.86 146.79 146.87 146.54
## 149 147.70 146.97 147.46 147.53 147.49 147.43 147.63 147.66 147.69 147.98
## 150 148.57 148.73 148.61 148.64 148.83 149.02 148.86 148.66 148.67 148.99
## 151 149.82 149.78 149.42 149.43 149.64 149.49 150.01 149.70 149.58 149.74
## 152 150.66 150.51 150.66 150.46 150.81 150.51 150.83 150.63 150.65 150.77
## 153 151.73 151.47 151.67 151.51 151.70 151.56 151.76 151.88 151.96 151.91
## 154 152.79 152.50 152.70 152.75 152.80 152.92 152.58 152.95 152.87 152.70
## 155 153.56 153.85 153.79 153.71 153.47 153.80 154.05 153.81 153.55 153.62
## 156 154.52 154.65 154.53 154.31 154.65 155.08 154.62 154.77 154.58 155.10
## 157 155.66 155.88 155.52 155.81 155.58 155.43 155.81 155.92 155.58 155.71
## 158 156.49 156.81 156.80 156.73 156.76 156.55 156.74 156.60 156.97 156.74
## 159 157.76 157.33 157.84 157.70 157.64 157.81 157.42 157.34 157.77 157.75
## 160 158.22 158.23 158.55 158.70 158.83 159.05 158.85 158.56 158.65 158.53
## 91 92 93 94 95 96 97 98 99 100
## 1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
## 2 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
## 3 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
## 4 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00
## 5 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
## 6 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00
## 7 7.00 7.00 7.00 7.00 7.00 7.00 7.00 7.00 7.00 7.00
## 8 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00
## 9 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00
## 10 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00
## 11 11.00 11.00 11.00 11.00 11.00 11.00 11.00 11.00 11.00 11.00
## 12 12.00 12.00 12.00 12.00 12.00 12.00 12.00 12.00 12.00 12.00
## 13 13.00 12.99 13.00 13.00 13.00 13.00 13.00 13.00 13.00 13.00
## 14 14.00 14.00 14.00 13.99 13.99 13.99 14.00 14.00 14.00 14.00
## 15 14.99 15.00 14.99 15.00 15.00 15.00 15.00 15.00 15.00 14.99
## 16 16.00 16.00 16.00 15.99 16.00 15.97 16.00 16.00 16.00 16.00
## 17 17.00 16.98 17.00 16.99 16.99 16.97 16.98 16.99 16.98 16.98
## 18 17.97 18.00 18.00 18.00 17.99 17.99 18.00 17.99 17.99 18.00
## 19 19.00 19.00 19.00 19.00 18.99 19.00 19.00 18.99 19.00 18.98
## 20 19.98 19.98 20.00 19.98 19.98 20.00 19.99 19.99 19.99 19.99
## 21 20.98 21.00 20.98 20.98 20.95 20.98 21.00 20.99 21.00 21.00
## 22 21.99 21.97 21.95 21.99 22.00 21.98 22.00 22.00 22.00 21.99
## 23 22.99 23.00 22.98 23.00 23.00 22.98 22.97 22.97 22.99 22.98
## 24 23.98 23.98 23.98 23.99 23.99 23.99 23.97 24.00 24.00 23.97
## 25 24.93 24.98 24.97 24.97 24.95 24.96 25.00 24.98 24.98 24.95
## 26 25.99 25.98 25.98 25.98 25.97 25.99 25.97 25.97 25.98 25.96
## 27 26.97 26.97 26.99 26.96 26.97 26.97 26.99 26.98 26.96 26.97
## 28 27.93 27.98 27.97 27.99 27.95 27.95 27.97 27.97 27.98 27.99
## 29 28.97 28.93 28.99 28.93 28.96 28.95 28.93 28.96 28.96 28.97
## 30 29.96 29.95 29.94 29.93 29.95 29.95 29.97 29.98 29.96 29.98
## 31 30.94 30.91 30.90 30.96 30.92 30.95 30.94 30.96 30.95 30.96
## 32 31.92 31.96 31.96 31.97 31.92 31.91 31.98 31.90 31.96 31.96
## 33 32.95 32.95 32.89 32.96 32.94 32.96 32.94 32.93 32.95 32.91
## 34 33.95 33.96 33.90 33.91 33.88 33.97 33.95 33.97 33.93 33.95
## 35 34.90 34.93 34.93 34.95 34.95 34.94 34.97 34.89 34.94 34.95
## 36 35.83 35.89 35.88 35.93 35.94 35.95 35.88 35.91 35.92 35.91
## 37 36.90 36.94 36.84 36.95 36.90 36.94 36.99 36.97 36.94 36.92
## 38 37.95 37.90 37.93 37.89 37.91 37.89 37.89 37.89 37.91 37.93
## 39 38.83 38.93 38.92 38.88 38.88 38.94 38.95 38.91 38.94 38.85
## 40 39.88 39.90 39.86 39.83 39.90 39.87 39.94 39.88 39.88 39.94
## 41 40.81 40.92 40.92 40.94 40.89 40.86 40.86 40.86 40.91 40.86
## 42 41.93 41.92 41.92 41.96 41.85 41.85 41.90 41.92 41.91 41.84
## 43 42.87 42.84 42.86 42.85 42.91 42.87 42.83 42.89 42.85 42.93
## 44 43.88 43.87 43.88 43.88 43.94 43.95 43.84 43.91 43.85 43.83
## 45 44.89 44.85 44.86 44.84 44.86 44.81 44.92 44.92 44.86 44.87
## 46 45.88 45.82 45.91 45.88 45.82 45.80 45.85 45.91 45.84 45.83
## 47 46.78 46.80 46.82 46.85 46.89 46.89 46.81 46.93 46.89 46.84
## 48 47.80 47.81 47.77 47.93 47.82 47.85 47.85 47.77 47.79 47.91
## 49 48.80 48.79 48.94 48.78 48.84 48.85 48.84 48.84 48.88 48.76
## 50 49.89 49.85 49.85 49.85 49.83 49.77 49.78 49.79 49.84 49.81
## 51 50.78 50.78 50.79 50.70 50.80 50.77 50.81 50.81 50.86 50.80
## 52 51.85 51.79 51.73 51.77 51.86 51.75 51.85 51.86 51.79 51.84
## 53 52.82 52.74 52.74 52.83 52.74 52.72 52.80 52.83 52.84 52.87
## 54 53.71 53.81 53.83 53.74 53.82 53.82 53.81 53.75 53.83 53.85
## 55 54.80 54.84 54.71 54.68 54.83 54.81 54.72 54.77 54.84 54.81
## 56 55.73 55.70 55.74 55.84 55.78 55.79 55.76 55.79 55.74 55.67
## 57 56.71 56.64 56.74 56.63 56.77 56.77 56.69 56.78 56.77 56.80
## 58 57.70 57.69 57.87 57.80 57.64 57.78 57.80 57.79 57.77 57.78
## 59 58.72 58.70 58.78 58.73 58.74 58.77 58.76 58.72 58.73 58.69
## 60 59.75 59.67 59.69 59.72 59.73 59.77 59.69 59.75 59.79 59.64
## 61 60.64 60.74 60.80 60.78 60.75 60.80 60.72 60.81 60.70 60.77
## 62 61.70 61.70 61.71 61.80 61.65 61.75 61.78 61.72 61.76 61.78
## 63 62.83 62.69 62.69 62.78 62.61 62.71 62.84 62.68 62.74 62.79
## 64 63.68 63.73 63.63 63.69 63.67 63.67 63.68 63.73 63.70 63.71
## 65 64.59 64.72 64.69 64.70 64.76 64.76 64.51 64.76 64.65 64.71
## 66 65.56 65.70 65.71 65.59 65.66 65.80 65.63 65.69 65.79 65.77
## 67 66.63 66.70 66.73 66.57 66.74 66.56 66.80 66.61 66.76 66.80
## 68 67.61 67.65 67.72 67.71 67.64 67.61 67.63 67.71 67.74 67.70
## 69 68.71 68.58 68.66 68.75 68.75 68.60 68.62 68.66 68.77 68.63
## 70 69.59 69.58 69.55 69.67 69.63 69.74 69.65 69.68 69.62 69.77
## 71 70.59 70.70 70.57 70.66 70.60 70.62 70.74 70.63 70.64 70.65
## 72 71.51 71.65 71.59 71.62 71.61 71.62 71.54 71.66 71.70 71.62
## 73 72.52 72.59 72.58 72.70 72.57 72.56 72.58 72.66 72.63 72.66
## 74 73.47 73.51 73.66 73.59 73.61 73.54 73.77 73.46 73.73 73.64
## 75 74.63 74.56 74.61 74.69 74.56 74.60 74.57 74.71 74.56 74.55
## 76 75.59 75.58 75.52 75.62 75.60 75.68 75.65 75.60 75.61 75.71
## 77 76.67 76.42 76.61 76.63 76.65 76.65 76.63 76.60 76.54 76.59
## 78 77.49 77.68 77.62 77.55 77.68 77.57 77.57 77.60 77.70 77.51
## 79 78.66 78.43 78.49 78.49 78.64 78.70 78.58 78.50 78.63 78.63
## 80 79.56 79.69 79.49 79.67 79.58 79.56 79.52 79.59 79.58 79.59
## 81 80.55 80.47 80.59 80.40 80.39 80.52 80.59 80.56 80.60 80.51
## 82 81.44 81.63 81.49 81.57 81.48 81.60 81.43 81.49 81.53 81.61
## 83 82.49 82.56 82.69 82.50 82.46 82.57 82.68 82.62 82.54 82.55
## 84 83.44 83.48 83.58 83.46 83.63 83.37 83.60 83.61 83.53 83.37
## 85 84.52 84.55 84.49 84.50 84.60 84.61 84.49 84.47 84.49 84.59
## 86 85.58 85.52 85.68 85.36 85.54 85.54 85.55 85.57 85.56 85.52
## 87 86.49 86.51 86.48 86.43 86.52 86.53 86.39 86.56 86.51 86.46
## 88 87.36 87.43 87.45 87.49 87.58 87.50 87.54 87.56 87.57 87.39
## 89 88.58 88.28 88.53 88.41 88.45 88.36 88.64 88.52 88.43 88.49
## 90 89.44 89.35 89.56 89.28 89.39 89.58 89.55 89.54 89.42 89.53
## 91 90.42 90.44 90.47 90.36 90.43 90.64 90.56 90.62 90.56 90.49
## 92 91.41 91.47 91.38 91.35 91.37 91.36 91.52 91.59 91.48 91.64
## 93 92.45 92.29 92.41 92.51 92.41 92.55 92.41 92.50 92.52 92.56
## 94 93.36 93.37 93.27 93.35 93.50 93.50 93.33 93.35 93.45 93.45
## 95 94.50 94.41 94.16 94.36 94.29 94.34 94.45 94.44 94.36 94.54
## 96 95.48 95.42 95.37 95.39 95.38 95.38 95.43 95.36 95.50 95.46
## 97 96.37 96.19 96.48 96.45 96.51 96.55 96.43 96.43 96.54 96.53
## 98 97.30 97.42 97.43 97.36 97.40 97.44 97.49 97.32 97.54 97.28
## 99 98.44 98.34 98.24 98.37 98.25 98.16 98.39 98.47 98.60 98.54
## 100 99.14 99.36 99.48 99.29 99.36 99.27 99.39 99.38 99.41 99.46
## 101 100.32 100.59 100.19 100.38 100.22 100.36 100.28 100.34 100.46 100.28
## 102 101.35 101.19 101.20 101.17 101.28 101.35 101.59 101.46 101.46 101.46
## 103 102.30 102.56 102.39 102.37 102.35 102.42 102.29 102.24 102.41 102.47
## 104 103.27 103.48 103.45 103.20 103.40 103.05 103.42 103.26 103.38 103.39
## 105 104.47 104.43 104.15 104.24 104.17 104.28 104.42 104.38 104.34 104.25
## 106 105.37 105.37 105.15 105.26 105.36 105.45 105.50 105.38 105.32 105.20
## 107 106.27 106.37 106.27 106.32 106.41 106.34 106.36 106.34 106.52 106.52
## 108 107.21 107.25 107.22 107.19 107.36 107.37 107.39 107.24 107.51 107.26
## 109 108.25 108.42 108.29 108.35 108.37 108.30 108.45 108.25 108.23 108.47
## 110 109.48 109.08 109.24 109.33 109.40 109.17 109.39 109.43 109.29 109.38
## 111 110.20 110.18 110.32 110.31 110.15 110.13 110.31 110.50 110.17 110.39
## 112 111.40 111.02 111.32 110.92 111.30 111.15 111.19 111.47 111.37 111.22
## 113 112.02 112.04 112.31 112.24 112.28 112.26 112.25 112.23 112.44 112.15
## 114 113.15 113.39 113.22 113.22 113.40 113.16 113.41 113.20 113.38 113.20
## 115 114.18 114.24 114.30 114.22 114.00 114.23 114.28 114.24 114.27 114.19
## 116 114.99 115.20 115.25 115.16 115.29 115.15 115.46 115.18 115.27 115.16
## 117 116.03 116.21 116.25 116.23 116.29 116.03 116.27 116.25 116.26 116.36
## 118 117.03 117.11 117.03 117.13 117.11 117.21 117.00 117.41 117.11 117.16
## 119 118.36 118.10 118.40 118.32 118.13 118.24 118.01 118.15 118.40 118.21
## 120 118.96 119.31 119.16 119.14 119.13 119.13 119.23 119.15 119.35 119.13
## 121 120.02 120.36 120.08 120.19 120.22 120.32 120.20 120.16 120.16 120.30
## 122 121.20 121.10 121.02 121.18 120.97 120.96 121.25 121.45 121.19 121.34
## 123 121.93 122.07 122.16 121.99 122.05 122.24 122.17 122.10 122.49 122.43
## 124 123.13 123.23 123.22 122.70 122.97 123.15 123.12 123.24 123.24 122.94
## 125 123.90 124.03 124.22 124.16 123.91 124.10 124.32 124.37 124.31 124.13
## 126 124.80 124.96 125.14 125.14 125.11 125.17 125.18 124.91 125.18 125.14
## 127 126.05 126.03 125.86 125.86 126.17 126.09 126.22 126.20 126.07 125.95
## 128 127.16 127.05 127.16 126.99 126.79 127.40 127.12 126.94 127.31 127.04
## 129 128.19 127.93 128.17 128.14 128.02 128.20 128.01 128.21 128.47 127.86
## 130 129.17 128.97 129.05 129.14 129.32 129.23 129.17 128.79 129.08 129.27
## 131 130.14 130.16 129.93 130.08 129.93 130.05 130.03 129.91 130.24 130.15
## 132 130.87 131.11 130.91 131.05 130.96 130.92 131.34 131.34 131.25 130.90
## 133 131.77 131.87 131.93 132.14 131.94 132.19 132.13 132.18 132.20 132.03
## 134 132.90 132.98 132.90 132.92 133.09 133.06 132.94 132.98 133.11 133.17
## 135 133.67 134.09 134.10 133.99 134.05 133.97 134.18 134.08 134.17 134.04
## 136 134.70 134.92 134.85 135.14 134.96 134.98 134.88 135.13 135.12 135.07
## 137 136.10 136.29 135.86 135.96 136.11 136.48 135.79 135.98 136.25 136.01
## 138 137.07 137.10 137.18 137.06 136.81 137.05 136.94 137.01 137.12 137.02
## 139 138.07 138.19 138.07 138.06 138.03 138.11 138.07 138.15 138.02 137.86
## 140 138.99 138.96 138.98 139.06 138.71 138.70 139.12 139.13 138.96 139.17
## 141 140.06 139.97 140.30 139.77 140.13 140.12 140.05 140.03 139.85 140.05
## 142 141.21 140.59 140.82 140.85 140.89 140.90 140.79 141.01 141.00 141.03
## 143 141.82 141.93 141.92 141.86 141.87 141.97 142.09 141.78 142.08 141.94
## 144 142.96 142.98 142.73 142.64 142.99 142.83 143.03 142.85 143.04 143.17
## 145 143.59 143.77 143.99 143.76 143.79 143.55 144.22 144.09 144.08 144.10
## 146 144.78 144.74 144.98 145.02 144.95 145.04 145.07 145.13 144.95 144.96
## 147 145.82 146.04 145.94 145.97 145.96 146.01 145.86 145.69 146.17 146.12
## 148 146.78 147.03 146.99 146.93 146.91 147.08 146.98 146.95 147.09 146.92
## 149 147.99 147.65 147.82 147.84 148.03 148.02 147.93 147.89 148.12 147.77
## 150 148.67 148.78 148.72 148.92 148.94 149.02 148.93 148.84 149.19 148.97
## 151 149.85 149.62 149.48 149.75 149.63 149.91 149.76 149.72 149.83 149.82
## 152 150.90 150.96 151.02 150.87 150.88 151.12 151.01 150.88 151.11 150.77
## 153 151.28 152.11 151.92 151.85 151.92 151.91 151.91 152.03 151.68 151.91
## 154 152.75 152.55 152.93 152.82 152.88 153.00 152.83 152.57 153.13 153.01
## 155 153.93 153.85 153.95 153.79 153.85 153.85 153.83 153.92 154.02 153.90
## 156 154.89 154.92 154.48 154.88 154.78 154.59 154.79 154.84 154.82 154.90
## 157 155.84 155.21 155.66 155.75 155.81 155.77 155.86 155.75 155.65 156.09
## 158 156.72 156.72 156.60 156.66 156.67 156.64 156.85 156.75 157.11 156.89
## 159 157.64 157.85 157.99 157.87 157.96 157.76 157.58 157.76 158.02 157.97
## 160 158.37 158.60 158.60 158.66 158.97 158.71 158.60 158.90 159.02 159.05Lorsque a et n sont très petits, je remarque que l’espérance de M est éloignée de la valeur de a, cela s’explique par le faible nombre d’essais réalisés pour trouver la valeur de a. Soit le nombre tiré en est proche, soit il en est drastiquement éloigné, ce qui fait baisser la moyenne. Lorsque a est petit et n grand, je remarque que l’espérance de M est proche de a. Lorsque n est petit et a grand, je remarque que l’espérance de M est éloignée de a, ce qui paraît normal car peu d’essais sont réalisés pour trouver la valeur de a. Or, comme c’est un nombre au hasard qui est choisi, ce nombre peut être éloigné de a. Lorsque a est grand et n est grand, je remarque que l’espérance de M est presque égale à a avec une distance de 1.5 au maximum.
Une formule pour E[M] en fonction de a et n où \(M = max_i X_i\) est \[E[M] = \sum_{i=0}^m \frac{X_i = a}{m}\] Je propose de mofidier M de la sorte : \(M = max_i (X_i = a)\) \[E[M] = \sum_{i=0}^m \frac{max_i (X_n = a)}{m}\] La valeur de cette espérance s’explique de cette façon : soit n le nombre d’essais avant d’obtenir la valeur a, \(X_n\) est donc égal à a. m est le nombre de fois que l’essai avec valeur a et n a été réalisé. Dans ce cas, l’espérance de M est égale à a.
Une formule pour trouver la variance de M est \[V[M] = E((X_n = a) - E((X_n = a)^2))\] Lorsque a et n sont très petits, je suppose que la variance est élevée, car l’espérance de M est plus proche de \(\frac{a}{2}\) que de a. Lorsque a est petit et n grand, je suppose que la variance est très faible, car généralement E[M] = a dans ces conditions. Lorsque a et n sont grands, je suppose que la variance est très faible, car généralement E[M] est très proche de a. En revanche, lorsque a est grand et n petit, je suppose que la variance est très élevée, car E[M] est plus proche de \(\frac{a}{2}\) que de a.
Mon intuition de départ (Si n >> a, E[M] = a) n’était donc pas si mauvaise, mais était incomplète, car lorsque a et n sont grands, E[M] se rapproche fortement de a.
###Q1.2
\(M' = \frac{2}{n} \sum_{i} X_i\) contient \(\sum_{i} \frac{X_i}{n}\), ce qui correspond à la moyenne des \(X_i\). La moyenne d’un ensemble de valeurs prises entre 1 et a est généralement très proche de \(\frac{a}{2}\). Multiplier par 2 cette valeur permettrait donc d’obtenir a. Je suppose que l’espérance de M’ est a (ou en tout cas très proche de a).
Tout comme pour la question précédente, j’ai étudié l’espérance de \(M' = \frac{2}{n} \sum_{i} X_i\) pour a pris dans [1, 160] et n choisi dans [1, 100].
set.seed(42)
a = 160
n = 100
#library(dplyr)
dm <- data.frame(x = 1:a)
for(i in 1:a) {
for(j in 1:n) {
S = replicate(100, 2*sum((sample(0:i, j, T))/j))
dm[i:i, j:j] = mean(S)
}
}
colnames(dm) <- 1:n #Pour nommer les colonnes avec les différentes valeurs prises par n
rownames(dm) <- 1:a #Pour nommer les lignes avec les différentes valeurs prises par a
#dmJ’observe qu’effectivement E[M’] a une valeur proche de a et permet donc de l’estimer. Lorsque a et n sont grands, cet estimateur donne une valeur très proche de a. Lorsque a est petit est n grand, E[M’] est presque égale à a (à 0.1 près). Lorsque a et n sont très petits, les valeurs de E[M’] sont globalement proches de a, ce qui est normal, car il y a moins de valeurs tirées dans l’échantillon pour déterminer a. Lorsque n est petit et a grand, selon l’échantillon sur lequel on réalise nos données, les valeurs sont globalement bien moins proches de a que lorsque n est grand (on peut même trouver une différence de plus de 10 entre a et la moyenne obtenue !)
Exercice 2 : Un deuxième jeu à base de max
Le but de Bob est de donner un nombre supérieur à M tout en restant inférieur à A et à 1, car A est pris uniformément dans l’intervalle [0, 1].
###Q2.3 Bob n’a pas encore trop pu réfléchir à la bonne stratégie et il se propose donc d’essayer la stratégie \(r(M) = 1.1M\) lorsque \(n=10\)
Avec cette stratégie, Bob est certain de donner une valeur plus grande que M, ce qui est souhaitable puisque M < A pour tout M et pour tout A. En revanche, Bob s’interdit quelques valeurs dans l’intervalle [0, 1] : en calculant \(M = \frac{1}{1.1}\), on obtient \(M \simeq 0.909\), ce qui signifie qu’avec sa méthode, la valeur qu’il obtient pour \(M > 0.909\) n’est pas dans l’intervalle [0, A], car \(A \leq 1\). En appliquant cette méthode, Alice gagne dans environ 10% des cas (lorsque \(M \simeq 0.909\)), je pense donc que cette stratégie m’a l’air bonne et vaut la peine d’être essayée par Bob.
set.seed(42)
n = 10
A = runif(1,0,1)
#On tire les valeurs de X_i uniformément dans l'intervalle [0, A]
M <- c(replicate(100, max(runif(n, 0, A))))
reponse = c(1:100)
EcartAvecA = c(1:100)
Gain = c(1:100)
dm23test = data.frame(numEssai = c(1:100), M, reponse, EcartAvecA, Gain)
#Calcul de r(M) que va annoncer Bob
#ici, r(M) est modélisé par reponse
for(i in 1:100) {
dm23test[i, 3] = 1.1*dm23test[i, 2]
}
#Calcul de l'écart avec A
for(j in 1:100) {
dm23test[j, "EcartAvecA"] = A - dm23test[j, 3]
}
#Calcul des gains de Bob : en effet, soit il gagne de l'argent, soit il ne gagne rien
for(k in 1:100) {
dm23test[k, 5] = 0
if(dm23test[k, "EcartAvecA"] >= 0) {
dm23test[k, 5] = dm23test[k, "EcartAvecA"]
}
}
#dm23test[1, 5] = 0
#dm23test[1, 5] = dm23test[dm23test[1, 4] >= 0,]
#dm23test[2, 5] = 0
#dm23test[2, 5] = dm23test[dm23test[1, 4] >= 0,]
dm23test## numEssai M reponse EcartAvecA Gain
## 1 1 0.8572423 0.9429665 -2.816043e-02 0.000000e+00
## 2 2 0.8948874 0.9843762 -6.957015e-02 0.000000e+00
## 3 3 0.9046441 0.9951085 -8.030250e-02 0.000000e+00
## 4 4 0.8293644 0.9123009 2.505152e-03 2.505152e-03
## 5 5 0.8906002 0.9796602 -6.485417e-02 0.000000e+00
## 6 6 0.7178417 0.7896259 1.251802e-01 1.251802e-01
## 7 7 0.8990871 0.9889958 -7.418978e-02 0.000000e+00
## 8 8 0.6580711 0.7238782 1.909279e-01 1.909279e-01
## 9 9 0.7097279 0.7807007 1.341054e-01 1.341054e-01
## 10 10 0.8535453 0.9388998 -2.409374e-02 0.000000e+00
## 11 11 0.8805996 0.9686596 -5.385355e-02 0.000000e+00
## 12 12 0.7655111 0.8420622 7.274381e-02 7.274381e-02
## 13 13 0.8805651 0.9686216 -5.381559e-02 0.000000e+00
## 14 14 0.8575262 0.9432789 -2.847282e-02 0.000000e+00
## 15 15 0.8642359 0.9506594 -3.585341e-02 0.000000e+00
## 16 16 0.8551817 0.9406999 -2.589381e-02 0.000000e+00
## 17 17 0.6675615 0.7343177 1.804884e-01 1.804884e-01
## 18 18 0.8398436 0.9238279 -9.021871e-03 0.000000e+00
## 19 19 0.8397042 0.9236746 -8.868530e-03 0.000000e+00
## 20 20 0.8097110 0.8906821 2.412393e-02 2.412393e-02
## 21 21 0.8852886 0.9738175 -5.901142e-02 0.000000e+00
## 22 22 0.8166644 0.8983308 1.647524e-02 1.647524e-02
## 23 23 0.8795565 0.9675121 -5.270606e-02 0.000000e+00
## 24 24 0.9116524 1.0028177 -8.801161e-02 0.000000e+00
## 25 25 0.8604344 0.9464779 -3.167183e-02 0.000000e+00
## 26 26 0.7307108 0.8037818 1.110242e-01 1.110242e-01
## 27 27 0.8651679 0.9516847 -3.687865e-02 0.000000e+00
## 28 28 0.8098056 0.8907861 2.401993e-02 2.401993e-02
## 29 29 0.8989428 0.9888370 -7.403100e-02 0.000000e+00
## 30 30 0.6925590 0.7618149 1.529911e-01 1.529911e-01
## 31 31 0.8513230 0.9364553 -2.164923e-02 0.000000e+00
## 32 32 0.9063860 0.9970246 -8.221853e-02 0.000000e+00
## 33 33 0.8554713 0.9410184 -2.621235e-02 0.000000e+00
## 34 34 0.7453452 0.8198797 9.492636e-02 9.492636e-02
## 35 35 0.8731222 0.9604344 -4.562839e-02 0.000000e+00
## 36 36 0.8704516 0.9574968 -4.269074e-02 0.000000e+00
## 37 37 0.8964254 0.9860679 -7.126188e-02 0.000000e+00
## 38 38 0.8841394 0.9725533 -5.774724e-02 0.000000e+00
## 39 39 0.8815890 0.9697479 -5.494186e-02 0.000000e+00
## 40 40 0.8830216 0.9713238 -5.651777e-02 0.000000e+00
## 41 41 0.8967649 0.9864414 -7.163532e-02 0.000000e+00
## 42 42 0.7116151 0.7827766 1.320295e-01 1.320295e-01
## 43 43 0.8957539 0.9853293 -7.052326e-02 0.000000e+00
## 44 44 0.8508919 0.9359811 -2.117507e-02 0.000000e+00
## 45 45 0.8428380 0.9271219 -1.231581e-02 0.000000e+00
## 46 46 0.8181965 0.9000162 1.478987e-02 1.478987e-02
## 47 47 0.7885866 0.8674453 4.736077e-02 4.736077e-02
## 48 48 0.8706424 0.9577066 -4.290056e-02 0.000000e+00
## 49 49 0.8963415 0.9859756 -7.116958e-02 0.000000e+00
## 50 50 0.7733707 0.8507077 6.409832e-02 6.409832e-02
## 51 51 0.8258922 0.9084814 6.324625e-03 6.324625e-03
## 52 52 0.7397138 0.8136852 1.011209e-01 1.011209e-01
## 53 53 0.8537613 0.9391375 -2.433141e-02 0.000000e+00
## 54 54 0.7244778 0.7969256 1.178805e-01 1.178805e-01
## 55 55 0.8794429 0.9673872 -5.258116e-02 0.000000e+00
## 56 56 0.8383730 0.9222103 -7.404304e-03 0.000000e+00
## 57 57 0.6947446 0.7642191 1.505870e-01 1.505870e-01
## 58 58 0.9084438 0.9992882 -8.448216e-02 0.000000e+00
## 59 59 0.9085120 0.9993632 -8.455720e-02 0.000000e+00
## 60 60 0.9111397 1.0022537 -8.744767e-02 0.000000e+00
## 61 61 0.7102967 0.7813263 1.334797e-01 1.334797e-01
## 62 62 0.8433722 0.9277094 -1.290339e-02 0.000000e+00
## 63 63 0.6714948 0.7386443 1.761617e-01 1.761617e-01
## 64 64 0.7795889 0.8575478 5.725828e-02 5.725828e-02
## 65 65 0.8826087 0.9708695 -5.606350e-02 0.000000e+00
## 66 66 0.8735510 0.9609061 -4.610003e-02 0.000000e+00
## 67 67 0.8243079 0.9067387 8.067307e-03 8.067307e-03
## 68 68 0.7143213 0.7857534 1.290526e-01 1.290526e-01
## 69 69 0.8738099 0.9611909 -4.638490e-02 0.000000e+00
## 70 70 0.8649100 0.9514010 -3.659495e-02 0.000000e+00
## 71 71 0.8073086 0.8880394 2.676660e-02 2.676660e-02
## 72 72 0.8023795 0.8826174 3.218863e-02 3.218863e-02
## 73 73 0.7457361 0.8203097 9.449636e-02 9.449636e-02
## 74 74 0.8880644 0.9768708 -6.206480e-02 0.000000e+00
## 75 75 0.8614678 0.9476145 -3.280850e-02 0.000000e+00
## 76 76 0.9077490 0.9985239 -8.371790e-02 0.000000e+00
## 77 77 0.8393445 0.9232790 -8.472907e-03 0.000000e+00
## 78 78 0.8708238 0.9579062 -4.310017e-02 0.000000e+00
## 79 79 0.7900017 0.8690019 4.580413e-02 4.580413e-02
## 80 80 0.8316022 0.9147624 4.366676e-05 4.366676e-05
## 81 81 0.8232567 0.9055823 9.223722e-03 9.223722e-03
## 82 82 0.8508987 0.9359885 -2.118247e-02 0.000000e+00
## 83 83 0.8808143 0.9688957 -5.408970e-02 0.000000e+00
## 84 84 0.8560812 0.9416893 -2.688326e-02 0.000000e+00
## 85 85 0.7435591 0.8179151 9.689099e-02 9.689099e-02
## 86 86 0.8949572 0.9844529 -6.964683e-02 0.000000e+00
## 87 87 0.7815874 0.8597461 5.505992e-02 5.505992e-02
## 88 88 0.8465066 0.9311573 -1.635121e-02 0.000000e+00
## 89 89 0.8448659 0.9293525 -1.454643e-02 0.000000e+00
## 90 90 0.8365726 0.9202299 -5.423852e-03 0.000000e+00
## 91 91 0.8644459 0.9508905 -3.608445e-02 0.000000e+00
## 92 92 0.9134254 1.0047680 -8.996195e-02 0.000000e+00
## 93 93 0.8983842 0.9882226 -7.341654e-02 0.000000e+00
## 94 94 0.8692515 0.9561766 -4.137059e-02 0.000000e+00
## 95 95 0.6918906 0.7610797 1.537264e-01 1.537264e-01
## 96 96 0.7340516 0.8074568 1.073492e-01 1.073492e-01
## 97 97 0.8242085 0.9066293 8.176700e-03 8.176700e-03
## 98 98 0.6523445 0.7175789 1.972271e-01 1.972271e-01
## 99 99 0.8114650 0.8926115 2.219455e-02 2.219455e-02
## 100 100 0.7852631 0.8637894 5.101660e-02 5.101660e-02EsperanceGainBob = mean(dm23test$Gain)
EsperanceGainBob## [1] 0.03167887#dm23testAvec cette stratégie, une estimation du gain de Bob est 0.0317 €, ce qui est très peu, de l’ordre de 3 centimes environ.
##Q2.4
On suppose que n = 3, ce qui va nous donner peu de valeurs, et donc des écarts plus grands entre nos valeurs, que si n avait été plus élevé. Comme on souhaite estimer la probabilité \(P[A=a, M=m]\) pour les différentes valeurs de \(a\) et de \(m\), \(\forall a\) et \(\forall m\) \(m \leq a\), je pense donc que l’on va se retrouver avec un tableau dont la motié inférieure gauche seulement sera remplie.
set.seed(42)
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionestimation <- function (a, m, n) {
A = sample(x = (0:10)/10, 1, T)
#A = as.integer(runif(1, 0, 10))/10
Xi = sample(x = (0:(A*10))/10, n, T)
M = max(Xi)
return (A==a && M == m)
}
dm24 <- data.frame(x = 1:11)
for(i in 0:10) {
for(j in 0:10) {
K = replicate(1000, estimation(i/10, j/10, 3))
dm24[i+1, j+1] = mean(K)
}
}
colnames(dm24) <- c("0", "0.1", "0.2", "0.3", "0.4", "0.5", "0.6", "0.7", "0.8", "0.9", "1.0")
rownames(dm24) <- c("0", "0.1", "0.2", "0.3", "0.4", "0.5", "0.6", "0.7", "0.8", "0.9", "1.0")
dm24## 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
## 0 0.091 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 0.1 0.010 0.083 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 0.2 0.001 0.034 0.059 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 0.3 0.002 0.006 0.027 0.048 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 0.4 0.000 0.006 0.011 0.036 0.035 0.000 0.000 0.000 0.000 0.000 0.000
## 0.5 0.000 0.001 0.008 0.015 0.033 0.040 0.000 0.000 0.000 0.000 0.000
## 0.6 0.001 0.001 0.005 0.009 0.019 0.024 0.044 0.000 0.000 0.000 0.000
## 0.7 0.000 0.000 0.005 0.005 0.014 0.020 0.031 0.036 0.000 0.000 0.000
## 0.8 0.000 0.000 0.001 0.004 0.006 0.016 0.017 0.018 0.028 0.000 0.000
## 0.9 0.000 0.000 0.000 0.003 0.007 0.009 0.011 0.007 0.018 0.027 0.000
## 1.0 0.001 0.000 0.000 0.002 0.003 0.006 0.007 0.016 0.019 0.019 0.016#Calcul de la somme de la colonne M=0.5
sumM5 = sum(dm24[,"0.5"])
sumM5## [1] 0.115Mon hypothèse se retrouve confirmée par cette estimation de \(P[A=a, M=m]\). Les valeurs pour que cette probabilité soit vraie sont très faibles, mais cela paraît normal, car on se retrouve en présence de \(\frac{11^2}{2} = \frac{122}{2} = 66\) cas présents non nuls dans l’univers étudié.
Mon tableau ne contient que les probabilités \(P[A=a, M=m]\) et non les probabilités conditionnelles \(P[A=a|M=m]\). Je cherche donc \(P[M=0.5]\) pour pouvoir calculer \(P[A=a|M=0.5]\) à l’aide de la formule de Bayes : \(P[A=a|M=m] = \frac{P[A=a, M=m]}{P[M=m]}\) D’après mon tableau, la somme des probabilité de la colonne \(M = 0.5\) me donnera la probabilité \(P[M = 0.5]\), soit \(P[M = 0.5] = 0.115\) Voici donc les différentes probabilités \(P[A=a|M=0.5]\) en fonction de a :
dm242 <- data.frame(x = 1:1)
for(k in 0:5) {
dm242[k+1] = dm24[k+6, 6] /0.115
}
colnames(dm242) <- c("A=0.5", "A=0.6", "A=0.7", "A=0.8", "A=0.9", "A=1.0")
rownames(dm242) <- c("P[A=a|M=0.5]")
dm242## A=0.5 A=0.6 A=0.7 A=0.8 A=0.9 A=1.0
## P[A=a|M=0.5] 0.3478261 0.2086957 0.173913 0.1391304 0.07826087 0.05217391Ainsi, la probabilité \(P[A=a|M=0.5]\) est plus élevée lorsque A = 0.5, dans le cas où Alice m’indique que \(M= 0.5\), je proposerai donc à Bob \(A = 0.5\).
##Q2.5
Comme \(\alpha > 1\), et que M est compris entre 0 et 1, on est certains que \(M^\alpha < 1\). Lorsque \(\alpha\) est très petit, \(M^\alpha\) aura une différence assez conséquente avec M, surtout lorsque M est faible. Lorsque \(\alpha\) est pris grand, \(M^\alpha\) sera très proche de M. Ainsi, la stratégie de Bob me semble très intéressante, tout dépend donc des valeurs de \(\alpha\) et de \(M\).
set.seed(82)
testStrategie <- function(a, n, alpha) {
M = max(runif(n, 0, a))
rM = M^alpha
gain = 0
if(rM <= a) {
gain = rM - M
}
return (gain)
}
dm25 <- data.frame(x = (1:20)*0.05, c = 1:10, c=1:10, c=1:10)
for(i in 1:20) {
K = replicate(1000, testStrategie(i*0.05, 2, 0.7))
dm25[i, 2] = mean(K)
}
for(i in 1:20) {
K = replicate(1000, testStrategie(i*0.05, 2, 0.5))
dm25[i, 3] = mean(K)
}
for(i in 1:20) {
K = replicate(1000, testStrategie(i*0.05, 2, 0.3))
dm25[i, 4] = mean(K)
}
colnames(dm25) <- c("A", "Alpha_0.7", "Alpha_0.5", "Alpha_0.3")
rbind(dm25, c("Espérance Gain", sum(dm25[, 2])/10, sum(dm25[, 3])/10, sum(dm25[, 4])/10))## A Alpha_0.7 Alpha_0.5
## 1 0.05 0.00216429260876494 8.68501155885363e-05
## 2 0.1 0.0077613849326303 0.000905794709309382
## 3 0.15 0.013453000283464 0.00230990323242131
## 4 0.2 0.019081058759039 0.00653402463999193
## 5 0.25 0.0302449435564045 0.0102613111960511
## 6 0.3 0.0362431975316746 0.0136825565037969
## 7 0.35 0.0451001315970525 0.025943762346752
## 8 0.4 0.0515616663922266 0.0326315655831174
## 9 0.45 0.059010332077493 0.0475971197429946
## 10 0.5 0.0693892923481722 0.0582716001228276
## 11 0.55 0.0749606524739513 0.0757134132637294
## 12 0.6 0.0775943334153378 0.085744329888235
## 13 0.65 0.0830890209207868 0.0998971710697783
## 14 0.7 0.0864475940960337 0.114603907189097
## 15 0.75 0.088171927355756 0.131704226788239
## 16 0.8 0.0886359572819094 0.132283120952374
## 17 0.85 0.0870590772789292 0.138214586752719
## 18 0.9 0.0861762770642492 0.145620551616676
## 19 0.95 0.0816655493317851 0.142296544827779
## 20 1 0.0719200485100683 0.135701706279007
## 21 Espérance Gain 0.115972973781573 0.140000404682048
## Alpha_0.3
## 1 0
## 2 0
## 3 0.000111163185159347
## 4 0
## 5 0.000216066476941176
## 6 0.000837297959303467
## 7 0.00213512345784249
## 8 0.00352555762714029
## 9 0.00737901516888544
## 10 0.0150953368435164
## 11 0.0203254307701122
## 12 0.0452219446698108
## 13 0.0578636946644884
## 14 0.078911887790331
## 15 0.0993753134015861
## 16 0.131655276729518
## 17 0.166905164235406
## 18 0.19719401064558
## 19 0.200870370495764
## 20 0.201094093429763
## 21 0.122871674755115Parmi ces trois valeurs, \(\alpha = 0.5\) est la meilleure, quoique de peu. Les gains de Bob restent relativement faibles, de l’ordre de 14 centimes environ. Cela me semble correct que \(\alpha = 0.5\) soit la meilleure valeur. Cela s’explique par le fait qu’avec un \(\alpha\) grand commme \(\alpha = 0.7\), pour un M petit, r(M) sera très grand. En revanche, pour M très grand (0.9 par exemple), \(r(M) = M^\alpha = 0.9^{0.7} \simeq 0.93\), qui donne un gain très faible : \(r(M) - M \simeq 0.03 €\). En revanche, avec \(\alpha = 0.5\), les gains sont à peu près constants pour les valeurs de a que j’ai prises afin de simuler la stratégie de Bob.
Voici un graphique présentant les valeurs des gains de Bob en fonction de la valeur de \(\alpha\). Attention, ne pas se fier à la légende, mais bien au graphe en lui-même pour les courbes ! En bleu, on a \(\alpha = 0.7\), en rouge on a \(\alpha = 0.5\) et en vert, on a \(\alpha = 0.3\) :
library(ggplot2)
ggplot(data = dm25, aes(x=A, y=Gains)) + geom_point(aes(A, Alpha_0.7, colour = "red")) + geom_point(aes(A, Alpha_0.5, colour = "blue")) + geom_point(aes(A, Alpha_0.3, colour = "green"))
Je calcule donc l’espérance des gains de Bob pour différentes valeurs de \(\alpha\) appartenant à l’intervalle [0, 1] :
set.seed(82)
dm252 <- data.frame(x = (1:20)*0.05, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10)
exposant = 0.1
for(j in 2:11) {
for(i in 1:20) {
K = replicate(1000, testStrategie(i*0.05, 2, exposant))
dm252[i, j] = mean(K)
}
exposant = exposant + 0.1
}
dm25Esperance <- data.frame(x = (1:1), c=1:1, c=1:1, c=1:1, c=1:1, c=1:1, c=1:1, c=1:1, c=1:1, c=1:1, c=1:1)
colnames(dm25Esperance) <- c("Alpha", "0.1", "0.2", "0.3", "0.4", "0.5", "0.6", "0.7", "0.8", "0.9", "1.0")
colnames(dm252) <- c("A", "0.1", "0.2", "0.3", "0.4", "0.5", "0.6", "0.7", "0.8", "0.9", "1.0")
dm25Esperance[1, 1] = "Espérance Gains"
esperanceMax = 0;
for(k in 2:11) {
dm25Esperance[1, k] = sum(dm252[, k])/10
if(dm25Esperance[1, k] > esperanceMax) {
esperanceMax = dm25Esperance[1, k]
}
}
dm25Esperance## Alpha 0.1 0.2 0.3 0.4 0.5
## 1 Espérance Gains 0.0658453 0.09778749 0.1228717 0.1347628 0.1422636
## 0.6 0.7 0.8 0.9 1.0
## 1 0.1329891 0.1165137 0.08861872 0.05031686 5.53665e-17esperanceMax## [1] 0.1422636En dehors de ces valeurs, je n’ai pas réussi à trouver un meilleur \(\alpha\) que \(\alpha = 0.5\) pour \(n = 3\).
Voyons voir ce que ça donne pour n = 10 :
set.seed(82)
dm253 <- data.frame(x = (1:20)*0.05, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10, c=1:10)
exposant = 0.1
for(j in 2:11) {
for(i in 1:20) {
K = replicate(1000, testStrategie(i*0.05, 10, exposant))
dm253[i, j] = mean(K)
}
exposant = exposant + 0.1
}
dm252Esperance <- data.frame(x = (1:1), c=1:1, c=1:1, c=1:1, c=1:1, c=1:1, c=1:1, c=1:1, c=1:1, c=1:1, c=1:1)
colnames(dm252Esperance) <- c("Alpha", "0.1", "0.2", "0.3", "0.4", "0.5", "0.6", "0.7", "0.8", "0.9", "1.0")
colnames(dm253) <- c("A", "0.1", "0.2", "0.3", "0.4", "0.5", "0.6", "0.7", "0.8", "0.9", "1.0")
dm252Esperance[1, 1] = "Espérance Gains"
esperanceMax = 0;
valAlpha = 0.1
alphaMax = 0
for(k in 2:11) {
dm252Esperance[1, k] = sum(dm253[, k])/10
if(dm252Esperance[1, k] > esperanceMax) {
esperanceMax = dm252Esperance[1, k]
alphaMax = valAlpha
}
valAlpha = valAlpha + 0.1
}
#dm253
dm252Esperance## Alpha 0.1 0.2 0.3 0.4 0.5
## 1 Espérance Gains 0.009180168 0.01090168 0.01394588 0.01731526 0.02118235
## 0.6 0.7 0.8 0.9 1.0
## 1 0.02468453 0.02828933 0.0295925 0.0261205 4.352317e-17esperanceMax## [1] 0.0295925alphaMax## [1] 0.8Cette fois-ci, parmi les 3 valeurs pour \(n = 10\), c’est pour \(\alpha = 0.7\) que l’espérance de M est la plus élevée. Il existe un meilleur \(\alpha\) en dehors de ces valeurs : c’est \(\alpha = 0.8\).
##Exercice 3 : Un dernier jeu à bas de max et de pile ou face
###Q36
Bob se retrouve dans le cas du jeu “Pile ou Face”. Ainsi notre univers se résume de la façon suivante : \(\Omega =\) {P, F} avec X(\(\Omega\)) = {0, 1}. Comme la pièce est non biaisée, \(\forall k\) appartient à {P, F}, P({k}) = \(\frac{1}{2}\). Calculons l’espérance de X : \(E[X] = 1 * \frac{1}{2} + 0 * \frac{1}{2} = \frac{1}{2}\)
Je vais ensuite modéliser la situation afin de vérifier mon calcul :
set.seed(42)
#s est un échantillon de 12000 lancers de pièces
#P(X == 0) correspond à l'évènement "La pièce est retombée sur le côté Pile, donc Alice donne A_2 à Bob"
#P(X == 1) correspond à l'évènement "La pièce est retombée sur le côté Face, donc Alice donne A_1 à Bob"
s = sample(0:1, 12000, T)
#On fait la moyenne des valeurs obtenues, ce qui correspond à l'espérance
e = mean(s)
e## [1] 0.5018333La valeur obtenue par ma modélisation valide ma valeur théorique de l’espérance de X. Comme \(E[X] = \frac{1}{2}\), la probabilité que Bob gagne est \(\frac{1}{2}\).