Chi-Squared Distribution

• Chi-Squared Distribution ( $\alpha = \nu /2$ and $\beta = 2$, special gamma distribution): The continuous random variable $X$ has a chi-squared distribution, with $\nu$ degrees of freedom, if its density function is given by
  $$f(x;\nu)=\left\lbrace \begin{array}{lr} \frac{1}{2^{\nu/2}... .../2-1}e^{-x/2}, & x>0 \\ 0,& elsewhere\\ \end{array}\right.$$
  where $\nu$ is a positive integer
• The chi-squared distribution plays a vital role in statistical inference.
• Topics dealing with sampling distributions, analysis of variance and nonparametric statistics involve extensive use of the chi-squared distribution.
• Theorem 6.4: