Distributions Related to the Standard Normal Distribution
The standard normal distribution generates, via various transformations, a few other distributions such as the chi-square, t, Cauchy, and F distributions. These distributions are fundamental in testing statistical hypotheses, as we will see in Chapters 5, 6, and 8.
4.6.1. The Chi-Square Distribution
Let X1,Xn be independent N(0, 1)-distributed random variables, and let
Yn = £ X2. (4.30)
The distribution of Yn is called the chi-square distribution with n degrees of freedom and is denoted by x2 or x2(n). Its distribution and density functions
can be derived recursively, starting from the case n = 1:
Gi(y) = P[71 < y] = P [X < y] = P[-Vy < Xi < Vt]
= j f (x)dx = 2 j f (x)dx for y > 0,
Gi(y) = 0 for y < 0,
where f (x) is defined by (4.28); hence,
gi(y) = G 1(y) = f (Vу) /vy = ^ for У > 0
gi(y) = 0 for у < 0.
Thus, g1(y) is the density of the /2 distribution. The corresponding momentgenerating function is
It follows easily from (4.30) – (4.32) that the moment-generating and characteristic functions of the x2 distribution are
Г(a) = j xa 1 exp(-x)dx.
The result (4.34) can be proved by verifying that for t < 1/2, (4.33) is the moment-generating function of (4.34). The function (4.35) is called the Gamma
function. Note that
Г(1) = 1,Г(1/2) = ^Л, Г(а + 1) = аГ(а) for а> 0. (4.36)
Moreover, the expectation and variance of the хП distribution are
E [Y„ ] = n, var(Y„) = 2n. (4.37)