11.1 The ${\chi }^2$ Distribution

Parameters: $𝜽=\nu$, with $\nu >0$ (usually an integer) called the degrees of freedom.

We write $X\sim {\chi }_{\nu }^2$.

Other: The ${\chi }_{\nu }^2$ distribution is the $\mathrm{Gamma}(\nu /2,1/2)$ distribution.

Transformations: If $Z_1,\mathrm{\dots },Z_n$ are independent $𝖭(0,1)$ random variables, then

In Example 8.1.3 we showed that the sum of a $\mathrm{𝖦𝖺𝗆}({\alpha }_1,1)$ and a $\mathrm{𝖦𝖺𝗆}({\alpha }_2,1)$ random variable is a $\mathrm{𝖦𝖺𝗆}({\alpha }_1+{\alpha }_2,1)$ random variable. But since a $\mathrm{\Gamma }(\alpha ,\beta )$ variable is just a $\mathrm{𝖦𝖺𝗆}(\alpha ,1)$ rv divided by $\beta$, this convolution result holds for any $\beta$, i.e. the sum of a $\mathrm{𝖦𝖺𝗆}({\alpha }_1,\beta )$ and a $\mathrm{𝖦𝖺𝗆}({\alpha }_2,\beta )$ random variable is a $\mathrm{𝖦𝖺𝗆}({\alpha }_1+{\alpha }_2,\beta )$ rv.

In Example 4.2.4 and again in Example 4.4.5 we showed that each $Z_i^2\sim \mathrm{𝖦𝖺𝗆}(1/2,1/2)$. Hence $Z_1^2+\mathrm{\dots }+Z_n^2\sim \mathrm{𝖦𝖺𝗆}(n/2,1/2)$.

1. $𝖤\left[X\right]=\nu$,

2. $\mathrm{𝖵𝖺𝗋}\left[X\right]=2\nu$.

Usage: Used in statistics as the distribution of the sum of square deviations (SSD) of a normal sample from its mean.