91Ó°ÊÓ

ξ(θ)be a p.d.f. that is defined as follows for constants

α >0 andβ >0:

\(\xi \left( \theta \right) = \left\{ \begin{aligned}{l}\frac{{{\beta ^\alpha }}}{{\Gamma \left( \alpha \right)}}{\theta ^{ - \left( {\alpha + 1} \right)}}{e^{ - \beta /\theta }}\,\,\,for\,\,\theta > 0\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,for\,\,\theta \le 0\end{aligned} \right.\)

A distribution with this p.d.f. is called an inverse gamma

distribution.

a.

\(Let,\,y = 1/\theta .\,Then\,\theta = 1/y\,and\,d\theta = - dy/{y^2}\)

Hence,

\(\begin{aligned}\int_0^\infty {\xi \left( \theta \right)} d\theta = \int_0^\infty {\frac{{{\beta ^\alpha }}}{{\Gamma \left( \alpha \right)}}} {y^{\alpha - 1}}\exp \left( { - \beta y} \right)dy\\ = \frac{{{\beta ^\alpha }}}{{\Gamma \left( \alpha \right)}}\int_0^\infty {{y^{\alpha - 1}}\exp \left( { - \beta y} \right)dy} \\ = \frac{{{\beta ^\alpha }}}{{\Gamma \left( \alpha \right)}} \times \frac{{\Gamma \left( \alpha \right)}}{{{\beta ^\alpha }}}\\ = 1\end{aligned}\)

Here the integration over whole range is 1.

So this is an actual p.d.f.

b.

If an observation X has a normal distribution with a known value of the mean\(\mu \)and an unknown value of the variance\(\theta \), then p.d.f of X has the form:

\(f\left( {x|\theta } \right) \propto \frac{{\exp \left( { - \frac{{{{\left( {x - \mu } \right)}^2}}}{{2\theta }}} \right)}}{{{\theta ^{1/2}}}}\).

Also, the prior p.d.f of\(\theta \)has the form:

\(\xi \left( \theta \right) \propto {\theta ^{ - \left( {\alpha + 1} \right)}}\exp \left( { - \beta /\theta } \right)\).

Therefore, the posterior p.d.f\(\xi \left( {\theta |x} \right)\)has the form:

\(\begin{aligned}\xi \left( {\theta |x} \right) \propto \xi \left( \theta \right)f\left( {x|\theta } \right)\\ \propto {\theta ^{ - \left( {\alpha + 3/2} \right)}}\exp \left\{ { - \left( {\beta + \frac{1}{2}{{\left( {x - \mu } \right)}^2}} \right) \cdot \frac{1}{\theta }} \right\}\end{aligned}\).

Hence, the posterior p.d.f. of \(\theta \) has the same form as \(\xi \left( \theta \right)\) with \(\alpha \) replaced by \(\alpha + 1/2\) and \(\beta \) replaced by \(\beta + 1/2{\left( {x - \mu } \right)^2}\), Since this distribution will be the prior distribution of \(\theta \) for future observations, it follows that the posterior distribution after any number of observations will also belong to the same family of distribution.