t分布、カイ２乗分布、F分布

Γ＝ガンマ関数、γ＝不完全ガンマ関数、B＝不完全ベータ関数、I=正規化不完全ベータ関数

t分布

確率密度関数

累積分布関数

import numpy as np
from pylab import *
import math

vs = [1.0,2.0,5.0,100.0]
colors = ['b','g','r','c']
xs = np.linspace(-5,5,100)
counter = 0
for v in vs:
    print counter
    fs = []
    c = colors[counter]
    for x in xs:
        fs.append(math.gamma((v+1)/2)/(math.sqrt(v*math.pi)*math.gamma(v/2))*((1+x**2/v)**(-(v+1)/2)))
    plot(xs,fs,color=c)
    counter = counter+1

stringArray = []
for v in vs:
    stringArray.append('v={0}'.format(v))
legend(stringArray)
title('Student\'s t distribution')
show()

カイ２乗分布

確率密度関数

累積分布関数

import numpy as np
from pylab import *
import math

ks = [1.0,2.0,3.0,4.0,5.0]
colors = ['b','g','r','c','m']
xs = np.linspace(0,8,50)
counter = 0
for k in ks:
    print counter
    fs = []
    c = colors[counter]
    for x in xs:
        f = ((0.5)**(k/2))/(math.gamma(k/2))*x**(k/2-1)*math.e**(-x/2)
        fs.append(f)
    plot(xs,fs,color=c)
    counter = counter+1

stringArray = []
for k in ks:
    stringArray.append('k={0}'.format(k))
legend(stringArray)
title('Chi-squared distribution')
show()

F分布

確率密度関数

累積分布関数

import numpy as np
from pylab import *
import math

def beta(a,b):
    beta = math.gamma(a)*math.gamma(b)/math.gamma(a+b)
    return beta

ds = [(1.0,4.0),(6.0,28.0),(28.0,6.0)]
colors = ['b','g','r']
xs = np.linspace(0,4,50)
counter = 0
for d in ds:
    print counter
    fs = []
    c = colors[counter]
    for x in xs:
        d1 = d[0]
        d2 = d[1]
        f = 1/(beta(d1/2,d2/2))*(d1*x/(d1*x+d2))**(d1/2)*(1-(d1*x/(d1*x+d2)))**(d2/2)*(x**(-1))
        fs.append(f)
    plot(xs,fs,color=c)
    counter = counter+1

stringArray = []
for d in ds:
    stringArray.append('(d1,d2)={0}'.format(d))
legend(stringArray)
title('F distribution')
show()