Question

Assume thatX1, . . . , Xnfrom a random sample from the normal distribution with meanμand variance \({\sigma ^2}\). Show that \({\hat \sigma ^2}\)has the gamma distribution with parameters \(\frac{{\left( {n - 1} \right)}}{2}\)and\(\frac{n}{{\left( {2{\sigma ^2}} \right)}}\).

Short answer

\({\hat \sigma ^2}\) follows Gamma distribution with parameters \(\frac{{n - 1}}{2}\) and \(\frac{n}{{2{\sigma ^2}}}\).

Step-by-step solution

Given information

\({X_1},{X_2},...,{X_n}\) are normal random variables.

Determine the distribution of \({\hat \sigma ^2}\) 

Let \(U = \frac{{n{{\hat \sigma }^2}}}{{{\sigma ^2}}}\) follows \({\chi ^2}\)a distribution with degrees of freedom \(n - 1\).i.e,

\(U\)follows Gamma distribution with parameters \(\frac{{n - 1}}{2}\) and \(\frac{1}{2}\).

Let, \(c = \frac{{{\sigma ^2}}}{n}\)

Then,

\(\begin{align}U &= \frac{{n{{\hat \sigma }^2}}}{{{\sigma ^2}}}\\ &= \frac{1}{c}{{\hat \sigma }^2}\\ \Rightarrow cU &= {{\hat \sigma }^2}\end{align}\)

Now,\(cU\) follows Gamma distribution with parameters \(\frac{{n - 1}}{2}\) and \(\frac{n}{{2{\sigma ^2}}}\).

Hence,\({\hat \sigma ^2}\) follows Gamma distribution with parameters \(\frac{{n - 1}}{2}\) and \(\frac{n}{{2{\sigma ^2}}}\).

Hence, proved.