Introduction to the Mathematical and Statistical Foundations of Econometrics

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

n

Yn = £ X2. (4.30)

j=1

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]

4у 4у

= j f (x)dx = 2 j f (x)dx for y > 0,

-Vt о

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

VyV 2n

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

and

Г(a) = j xa 1 exp(-x)dx.

0

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)