(Pearson probability distribution) Distributions where the probability density functions f x(x) satisfy the differential equation

where

a, 6(), b u b2 are real constants, are called Pearson probability distributions.

(a) Show that if ixA = E(X - E(X))k, then

where A = 10|x4|jl2 - 1 8 ^ - 12|X5.

(b) Show that normal, gamma, and beta distributions belong to the family of Pearson probability distributions.

dfx(x) dx (x - a)fx(x) b+bx + bx2"