Let () be a p.d.f. that is defined as follows for constants >0 and >0: A distribution with this p.d.f. is called an

Let ξ(θ) be a p.d.f. that is defined as follows for constants α >0 and β >0:

A distribution with this p.d.f. is called an inverse gamma distribution.
a. Verify that ξ(θ) is actually a p.d.f. by verifying that
b. Consider the family of probability distributions that can be represented by a p.d.f. ξ(θ) having the given form for all possible pairs of constants α > 0 and β > 0. Show that this family is a conjugate family of prior distributions for samples from a normal distribution with a known value of the mean μ and an unknown value of the variance θ.

-d-(+1)e-f/0 for > 0, for < 0. (9)