91Ó°ÊÓ

Suppose that \({X_1},...,{X_n}\) are i.i.d. having the normal distribution with mean μ and precision \(\tau \) given (μ, \(\tau \) ).It is required to prove that the posterior distribution of \(V = \left( {n - 1} \right){\sigma '^{2}}\tau \) is the χ2 distribution with n − 1 degrees of freedom

Let \({X_1},...,{X_n}\) be iid random variables from distribution with unknown mean \(\mu \) and unknown variance \({\sigma ^2} > 0\) Suppose that for the joint prior distribution of \(\mu \) and \(\tau = \frac{1}{{{\sigma ^2}}}\) is from normal distribution with mean \({\mu _0}\) and precision \({\lambda _0}\tau \) and the marginal distribution is gamma distribution with parameters \({\alpha _0}\,\,\,\,{\rm{and}}\,\,\,{\beta _0}\) .

Then the prior distribution of \(\mu \) given \(\tau = \frac{1}{{{\sigma ^2}}}\)is from normal distribution with mean \({\mu _1}\) and \(\tau = \frac{1}{{{\sigma ^2}}}\)

Where \({\mu _1} = \frac{{{\lambda _0}{\mu _0} + n{{\bar x}_n}}}{{{\lambda _0} + n}}\) and \({\lambda _1} = {\lambda _0} + n\)

The marginal distribution of \(\tau \) is the gamma distribution with parameters \({\alpha _0} > 0,{\beta _0} > 0\) where

\(\begin{align}{\alpha _1} &= {\alpha _0} + \frac{n}{2},\\{\beta _1} &= {\beta _0} + \frac{1}{2}s_n^{2} + \frac{{n{\lambda _0}{{\left( {{{\bar x}_n} - {\mu _0}} \right)}^2}}}{{2\left( {{\lambda _0} + n} \right)}}\end{align}\)

It is to be noted that in \(V = \left( {n - 1} \right){\sigma '^2}\tau \)the random variable is \(\tau \) and the constant is \(\left( {n - 1} \right){\sigma '^2}\tau \) .The posterior distribution of \(\tau \) is the gamma distribution with parameters

\( - \frac{1}{2} + \frac{n}{2} = \frac{{n - 1}}{2}\,\,\,\,\,\,{\rm{and}}\,\,\,\,\frac{{s_n^2}}{2} + \frac{{n \times 0 \times {{\left( {\bar x - 0} \right)}^2}}}{{2\left( {0 + n} \right)}} = \frac{{s_n^2}}{2}\) ,by multiplying a gamma distribution with a constant we the distribution of V as the gamma distribution with parameters \(\frac{{n - 1}}{2}\,\,{\rm{and}}\,\,\,\frac{{s_n^2}}{2} \times \frac{1}{{\left( {n - 1} \right){{\sigma '}^2}}} = \frac{1}{2}\)

Hence the posterior distribution of \(V = \left( {n - 1} \right){\sigma '^2}\tau \) is the χ2 distribution with n − 1 degrees of freedom.