## 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) /...

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