Question: Many “named” distributions are special cases of the more common distributions already discussed. For each of the following named distributions derive the form of the pdf, verify that it is a pdf, and calculate the mean variance.

(a) If X ~ exponential (β), then has the Weibull(γ, β) distribution, where is constant.

(c) If X ~ gamma (a, b), then has the inverted gamma IG(a, b) distribution.

(This distribution is useful in Bayesion estimation of variances; see Exercise 7.23 below)