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Chapter 8: Q13 E (page 473)

Prove that the distribution of\({\hat \sigma _0}^2\)in Examples 8.2.1and 8.2.2 is the gamma distribution with parameters\(\frac{n}{2}\)and\(\frac{n}{{2{\sigma ^2}}}\).

Short Answer

Expert verified

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

Step by step solution

01

Given information

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

02

Calculate the probability 

Let \(Z = \frac{{n{{\hat \sigma }^2}}}{{{\sigma ^2}}}\)follows \({\chi ^2}\)a distribution with n degrees of freedom.

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

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

Then,

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

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

Hence, proved.

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