Many named distributions axe special cases of the more common distributions already discussed. For each of the

Question:

Many "named" distributions axe 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 and variance.
(a) If X ~ exponential(β), then Y = X1/γ has the Weibull(γ, β) where γ > 0 is a constant.
(b) If X ~ exponential(β), then Y = (2X/β)1/2 has the Rayleigh If X ~ gamma(a, b), then Y = 1/X has the inverted gamma IG(a, b) (This is useful in Bayesian estimation of variances; see Exercise 7.23.)
(d) If X ~ gamma(3/2, β), then Y = (X/β)1/2 has the Maxwell If X ~ exponential(l), then Y = a - γ log X has the Gumbel(a, γ) where -∞ < a < ∞ and γ > 0. (The Gumbel is also known as the extreme value
Distribution
The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most...