The Chi-Square Distribution

Introduces the chi-square distribution as the distribution of a sum of squared independent standard normal random variables. Covers the definition with degrees of freedom, the probability density function expressed via the gamma function, and the mean and variance formulas $E[X]=k$ and $\mathrm{Var}(X)=2k$ .

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Tutorial

Introduction: Defining the Chi-Square Distribution

Let $Z_1, Z_2, \ldots, Z_k$ be independent random variables, each following the standard normal distribution $N(0,1).$ The sum of their squares,

X = Z_1^2 + Z_2^2 + \cdots + Z_k^2,

follows a chi-square distribution with $k$ degrees of freedom, written $X \sim \chi^2(k).$ Here $k$ is a positive integer.

For example, if $Z_1, Z_2, Z_3$ are independent standard normal random variables, then

Z_1^2 + Z_2^2 + Z_3^2 \sim \chi^2(3).

Because $X$ is a sum of squares, $X \geq 0$ always. Therefore the chi-square distribution is supported on $[0,\infty).$