DISTRIBUCION JI CUADRADO Jorge M. Galbiati Funci´n de densidad: o f (x) = 1 xk/2?1 e?x/2 k/2 ?(k/2) 2 si x > 0
Espacio param´trico: e Valor esperado: Varianza: 2k k
Grados de libertad k ? {1, 2, 3, …}
o Funci´n 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 acumuxlada 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 tienedistribuci´n ji-cuadrado con k grados de libertad, o X?k o entonces si k es grande la variable aleatoria Z = ?(2k) tiene distribuci´n aproximada normal standard. En la pr´ctica, si k es grande, si se requiere la probabilidad acumulada F (x) con a F distribuci´n ji-cuadrado, se puede obtener su valor aproximado buscando en la o tabla normal x?k FN ? (2k) o a en que FN es la distribuci´n normal est´ndar. Sepuede utilizar, como criterio, la condici´n k > 200. o
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 7.4537.991 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 39.0442.76 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 1.3441.735 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 12.8813.56 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 91.57162.7 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 9.31210.09 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…