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