4.4.4 The Chi-Squared Distribution

The chi-squared distribution is important because it is the basis for a number of procedures in statistical inference. The central role played by the chi-squared distribution in inference springs from its relationship to normal distributions (see Exercise 71). We’ll discuss this distribution in more detail in later chapters.

chi-squared distribution

Let $\nu$ be a positive integer. Then a random variable $X$ is said to have a chi-squared distribution with parameter $\nu$ if the pdf of $X$ is the gamma density with $\alpha =\nu /2$ and $\beta =2$ . The pdf of a chi-squared rv is thus

$$f\left(x;\nu \right)=\{\begin{array}{cc}\frac{1}{2^{\nu /2}\Gamma \left(v/2\right)}{x}^{\left(\nu /2\right)-1}{e}^{-x/2}& x\ge 0\\ 0& x<0\end{array}\text{\hspace{1em}(4.10)}$$

The parameter $\nu$ is called the number of degrees of freedom (df) of $X$. The symbol ${\chi }^2$ is often used in place of “chi-squared”.