DISTRIBUCION JI CUADRADO Jorge M. Galbiati Función de densidad: f (x) = 1 xk/2−1 e−x/2 k/2 2 Γ(k/2) Grados de libertad k ∈ {1, 2, 3, ...} Espacio paramétrico: Valor esperado: Varianza: si x > 0 k 2k Función generadora de momentos: 1 1−2t k/2 para t < 1/2 TABLA DE DISTRIBUCION JI CUADRADO La tabla entrega valores de la cuantila z para valores dados de probabilidad acumux lada F (x) = 0 f (y)dy. Los valores de probabilidad acumulada son 0.0005; 0.001; 0.0025; 0.005; 0.010; 0.025; desde 0.050 hasta 0.950, variando en 0.050; 0.975; 0.990; 0.995; 0.998; 0.999; 0.9995. Los valores de los grados de libertad son: De 1 a 30; de 35 a 100 variando en 5; 110; 120; 200. f(y) F(x) 0 x y APROXIMACION NORMAL DE LA JI-CUADRADO. Si una variable aleatoria X tiene distribución ji-cuadrado con k grados de libertad, tiene distribución aproximada entonces si k es grande la variable aleatoria Z = √X−k (2k) normal standard. En la práctica, si k es grande, si se requiere la probabilidad acumulada F (x) con F distribución ji-cuadrado, se puede obtener su valor aproximado buscando en la tabla normal x−k FN √ (2k) en que FN es la distribución normal estándar. Se puede utilizar, como criterio, la condición k > 200. TABLA DE DISTRIBUCION JI - CUADRADO Grados de Libertad 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 110 120 200 0.0005 0.000 0.001 0.015 0.064 0.158 0.299 0.485 0.710 0.972 1.265 1.587 1.934 2.305 2.697 3.108 3.536 3.980 4.439 4.912 5.398 5.896 6.404 6.924 7.453 7.991 8.538 9.093 9.656 10.23 10.80 13.79 16.91 20.14 23.46 26.87 30.34 33.88 37.47 41.11 44.79 48.52 52.28 56.07 59.90 67.63 75.47 140.7 Probabilidad acumulada 0.001 0.000 0.002 0.024 0.091 0.210 0.381 0.598 0.857 1.152 1.479 1.834 2.214 2.617 3.041 3.483 3.942 4.416 4.905 5.407 5.921 6.447 6.983 7.529 8.085 8.649 9.222 9.803 10.39 10.99 11.59 14.69 17.92 21.25 24.67 28.17 31.74 35.36 39.04 42.76 46.52 50.32 54.16 58.02 61.92 69.79 77.76 143.8 0.0025 0.000 0.005 0.045 0.145 0.307 0.527 0.794 1.104 1.450 1.827 2.232 2.661 3.112 3.582 4.070 4.573 5.092 5.623 6.167 6.723 7.289 7.865 8.450 9.044 9.646 10.26 10.87 11.50 12.13 12.76 16.03 19.42 22.90 26.46 30.10 33.79 37.54 41.33 45.17 49.04 52.95 56.89 60.86 64.86 72.92 81.07 148.4 0.005 0.000 0.010 0.072 0.207 0.412 0.676 0.989 1.344 1.735 2.156 2.603 3.074 3.565 4.075 4.601 5.142 5.697 6.265 6.844 7.434 8.034 8.643 9.260 9.886 10.52 11.16 11.81 12.46 13.12 13.79 17.19 20.71 24.31 27.99 31.73 35.53 39.38 43.28 47.21 51.17 55.17 59.20 63.25 67.33 75.55 83.85 152.2 0.010 0.000 0.020 0.115 0.297 0.554 0.872 1.239 1.646 2.088 2.558 3.053 3.571 4.107 4.660 5.229 5.812 6.408 7.015 7.633 8.260 8.897 9.542 10.20 10.86 11.52 12.20 12.88 13.56 14.26 14.95 18.51 22.16 25.90 29.71 33.57 37.48 41.44 45.44 49.48 53.54 57.63 61.75 65.90 70.06 78.46 86.92 156.4 0.025 0.001 0.051 0.216 0.484 0.831 1.237 1.690 2.180 2.700 3.247 3.816 4.404 5.009 5.629 6.262 6.908 7.564 8.231 8.907 9.591 10.28 10.98 11.69 12.40 13.12 13.84 14.57 15.31 16.05 16.79 20.57 24.43 28.37 32.36 36.40 40.48 44.60 48.76 52.94 57.15 61.39 65.65 69.92 74.22 82.87 91.57 162.7 0.050 0.004 0.103 0.352 0.711 1.145 1.635 2.167 2.733 3.325 3.940 4.575 5.226 5.892 6.571 7.261 7.962 8.672 9.390 10.12 10.85 11.59 12.34 13.09 13.85 14.61 15.38 16.15 16.93 17.71 18.49 22.47 26.51 30.61 34.76 38.96 43.19 47.45 51.74 56.05 60.39 64.75 69.13 73.52 77.93 86.79 95.70 168.3 0.100 0.016 0.211 0.584 1.064 1.610 2.204 2.833 3.490 4.168 4.865 5.578 6.304 7.042 7.790 8.547 9.312 10.09 10.86 11.65 12.44 13.24 14.04 14.85 15.66 16.47 17.29 18.11 18.94 19.77 20.60 24.80 29.05 33.35 37.69 42.06 46.46 50.88 55.33 59.79 64.28 68.78 73.29 77.82 82.36 91.47 100.6 174.8 0.150 0.036 0.325 0.798 1.366 1.994 2.661 3.358 4.078 4.817 5.570 6.336 7.114 7.901 8.696 9.499 10.31 11.12 11.95 12.77 13.60 14.44 15.28 16.12 16.97 17.82 18.67 19.53 20.39 21.25 22.11 26.46 30.86 35.29 39.75 44.24 48.76 53.29 57.84 62.41 66.99 71.59 76.20 80.81 85.44 94.72 104.0 179.4 0.200 0.064 0.446 1.005 1.649 2.343 3.070 3.822 4.594 5.380 6.179 6.989 7.807 8.634 9.467 10.31 11.15 12.00 12.86 13.72 14.58 15.44 16.31 17.19 18.06 18.94 19.82 20.70 21.59 22.48 23.36 27.84 32.34 36.88 41.45 46.04 50.64 55.26 59.90 64.55 69.21 73.88 78.56 83.25 87.95 97.36 106.8 183.0 0.250 0.102 0.575 1.213 1.923 2.675 3.455 4.255 5.071 5.899 6.737 7.584 8.438 9.299 10.17 11.04 11.91 12.79 13.68 14.56 15.45 16.34 17.24 18.14 19.04 19.94 20.84 21.75 22.66 23.57 24.48 29.05 33.66 38.29 42.94 47.61 52.29 56.99 61.70 66.42 71.14 75.88 80.62 85.38 90.13 99.67 109.2 186.2 DISTRIBUCION JI - CUADRADO (2) Grados de Libertad Probabilidad acumulada 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 110 120 200 0.300 0.148 0.713 1.424 2.195 3.000 3.828 4.671 5.527 6.393 7.267 8.148 9.034 9.926 10.82 11.72 12.62 13.53 14.44 15.35 16.27 17.18 18.10 19.02 19.94 20.87 21.79 22.72 23.65 24.58 25.51 30.18 34.87 39.58 44.31 49.06 53.81 58.57 63.35 68.13 72.92 77.71 82.51 87.32 92.13 101.8 111.4 189.0 0.350 0.206 0.862 1.642 2.470 3.325 4.197 5.082 5.975 6.876 7.783 8.695 9.612 10.53 11.45 12.38 13.31 14.24 15.17 16.11 17.05 17.98 18.92 19.87 20.81 21.75 22.70 23.64 24.59 25.54 26.49 31.25 36.02 40.81 45.61 50.42 55.24 60.07 64.90 69.74 74.58 79.43 84.29 89.14 94.00 103.7 113.5 191.7 0.400 0.275 1.022 1.869 2.753 3.655 4.570 5.493 6.423 7.357 8.295 9.237 10.18 11.13 12.08 13.03 13.98 14.94 15.89 16.85 17.81 18.77 19.73 20.69 21.65 22.62 23.58 24.54 25.51 26.48 27.44 32.28 37.13 42.00 46.86 51.74 56.62 61.51 66.40 71.29 76.19 81.09 85.99 90.90 95.81 105.6 115.5 194.3 0.450 0.357 1.196 2.109 3.047 3.996 4.952 5.913 6.877 7.843 8.812 9.783 10.76 11.73 12.70 13.68 14.66 15.63 16.61 17.59 18.57 19.55 20.53 21.51 22.49 23.47 24.45 25.44 26.42 27.40 28.39 33.31 38.23 43.16 48.10 53.04 57.98 62.92 67.87 72.81 77.76 82.71 87.67 92.62 97.57 107.5 117.4 196.8 0.500 0.455 1.386 2.366 3.357 4.351 5.348 6.346 7.344 8.343 9.342 10.34 11.34 12.34 13.34 14.34 15.34 16.34 17.34 18.34 19.34 20.34 21.34 22.34 23.34 24.34 25.34 26.34 27.34 28.34 29.34 34.34 39.34 44.34 49.33 54.33 59.33 64.33 69.33 74.33 79.33 84.33 89.33 94.33 99.33 109.3 119.3 199.3 0.550 0.571 1.597 2.643 3.687 4.728 5.765 6.800 7.833 8.863 9.892 10.92 11.95 12.97 14.00 15.02 16.04 17.06 18.09 19.11 20.13 21.15 22.17 23.19 24.20 25.22 26.24 27.26 28.27 29.29 30.31 35.39 40.46 45.53 50.59 55.65 60.71 65.77 70.82 75.88 80.93 85.98 91.02 96.07 101.1 111.2 121.3 201.9 0.600 0.708 1.833 2.946 4.045 5.132 6.211 7.283 8.351 9.414 10.47 11.53 12.58 13.64 14.69 15.73 16.78 17.82 18.87 19.91 20.95 21.99 23.03 24.07 25.11 26.14 27.18 28.21 29.25 30.28 31.32 36.47 41.62 46.76 51.89 57.02 62.13 67.25 72.36 77.46 82.57 87.67 92.76 97.85 102.9 113.1 123.3 204.4 0.650 0.873 2.100 3.283 4.438 5.573 6.695 7.806 8.909 10.01 11.10 12.18 13.27 14.35 15.42 16.49 17.56 18.63 19.70 20.76 21.83 22.89 23.95 25.01 26.06 27.12 28.17 29.23 30.28 31.33 32.38 37.62 42.85 48.06 53.26 58.45 63.63 68.80 73.97 79.13 84.28 89.43 94.58 99.72 104.9 115.1 125.4 207.1 0.700 1.074 2.408 3.665 4.878 6.064 7.231 8.383 9.524 10.66 11.78 12.90 14.01 15.12 16.22 17.32 18.42 19.51 20.60 21.69 22.77 23.86 24.94 26.02 27.10 28.17 29.25 30.32 31.39 32.46 33.53 38.86 44.16 49.45 54.72 59.98 65.23 70.46 75.69 80.91 86.12 91.32 96.52 101.7 106.9 117.3 127.6 210.0 0.750 1.323 2.773 4.108 5.385 6.626 7.841 9.037 10.22 11.39 12.55 13.70 14.85 15.98 17.12 18.25 19.37 20.49 21.60 22.72 23.83 24.93 26.04 27.14 28.24 29.34 30.43 31.53 32.62 33.71 34.80 40.22 45.62 50.98 56.33 61.66 66.98 72.28 77.58 82.86 88.13 93.39 98.65 103.9 109.1 119.6 130.1 213.1 DISTRIBUCION JI - CUADRADO (3) Grados de Libertad Probabilidad acumulada 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 110 120 200 0.800 1.642 3.219 4.642 5.989 7.289 8.558 9.803 11.03 12.24 13.44 14.63 15.81 16.98 18.15 19.31 20.47 21.61 22.76 23.90 25.04 26.17 27.30 28.43 29.55 30.68 31.79 32.91 34.03 35.14 36.25 41.78 47.27 52.73 58.16 63.58 68.97 74.35 79.71 85.07 90.41 95.73 101.1 106.4 111.7 122.2 132.8 216.6 0.850 2.072 3.794 5.317 6.745 8.115 9.446 10.75 12.03 13.29 14.53 15.77 16.99 18.20 19.41 20.60 21.79 22.98 24.16 25.33 26.50 27.66 28.82 29.98 31.13 32.28 33.43 34.57 35.71 36.85 37.99 43.64 49.24 54.81 60.35 65.86 71.34 76.81 82.26 87.69 93.11 98.51 103.9 109.3 114.7 125.4 136.1 220.7 0.900 2.706 4.606 6.251 7.779 9.236 10.64 12.02 13.36 14.68 15.99 17.28 18.55 19.81 21.06 22.31 23.54 24.77 25.99 27.20 28.41 29.62 30.81 32.01 33.20 34.38 35.56 36.74 37.92 39.09 40.26 46.06 51.81 57.51 63.17 68.80 74.40 79.97 85.53 91.06 96.58 102.1 107.6 113.0 118.5 129.4 140.2 226.0 0.950 3.841 5.992 7.815 9.488 11.07 12.59 14.07 15.51 16.92 18.31 19.68 21.03 22.36 23.68 25.00 26.30 27.59 28.87 30.14 31.41 32.67 33.92 35.17 36.42 37.65 38.89 40.11 41.34 42.56 43.77 49.80 55.76 61.66 67.50 73.31 79.08 84.82 90.53 96.22 101.9 107.5 113.1 118.8 124.3 135.5 146.6 234.0 0.975 5.024 7.379 9.348 11.14 12.83 14.45 16.01 17.53 19.02 20.48 21.92 23.34 24.74 26.12 27.49 28.85 30.19 31.53 32.85 34.17 35.48 36.78 38.08 39.36 40.65 41.92 43.19 44.46 45.72 46.98 53.20 59.34 65.41 71.42 77.38 83.30 89.18 95.02 100.8 106.6 112.4 118.1 123.9 129.6 140.9 152.2 241.1 0.990 6.635 9.214 11.34 13.28 15.09 16.81 18.48 20.09 21.67 23.21 24.72 26.22 27.69 29.14 30.58 32.00 33.41 34.81 36.19 37.57 38.93 40.29 41.64 42.98 44.31 45.64 46.96 48.28 49.59 50.89 57.34 63.69 69.96 76.15 82.29 88.38 94.42 100.4 106.4 112.3 118.2 124.1 130.0 135.8 147.4 159.0 249.4 0.995 7.879 10.60 12.84 14.86 16.75 18.55 20.28 21.95 23.59 25.19 26.76 28.30 29.82 31.32 32.80 34.27 35.72 37.16 38.58 40.00 41.40 42.80 44.18 45.56 46.93 48.29 49.64 50.99 52.34 53.67 60.27 66.77 73.17 79.49 85.75 91.95 98.11 104.2 110.3 116.3 122.3 128.3 134.2 140.2 151.9 163.6 255.3 0.998 9.141 12.00 14.32 16.42 18.39 20.25 22.04 23.77 25.46 27.11 28.73 30.32 31.88 33.43 34.95 36.46 37.95 39.42 40.88 42.34 43.78 45.20 46.62 48.03 49.44 50.83 52.22 53.59 54.97 56.33 63.08 69.70 76.22 82.66 89.03 95.34 101.6 107.8 114.0 120.1 126.2 132.3 138.3 144.3 156.2 168.1 260.7 0.999 10.83 13.85 16.27 18.47 20.52 22.46 24.32 26.12 27.88 29.59 31.26 32.91 34.53 36.12 37.70 39.25 40.79 42.31 43.82 45.31 46.80 48.27 49.73 51.18 52.62 54.05 55.48 56.89 58.30 59.70 66.62 73.40 80.08 86.66 93.17 99.61 106.0 112.3 118.6 124.8 131.0 137.2 143.3 149.4 161.6 173.6 267.5 0.9995 12.12 15.27 17.73 20.00 22.11 24.10 26.02 27.87 29.67 31.42 33.14 34.82 36.48 38.11 39.72 41.31 42.88 44.43 45.97 47.50 49.01 50.51 52.00 53.48 54.95 56.41 57.86 59.30 60.73 62.16 69.20 76.09 82.88 89.56 96.16 102.7 109.2 115.6 121.9 128.3 134.5 140.8 147.0 153.2 165.4 177.6 272.4