For a 0 and b 0 we say that a random variable has a beta d

For a > 0 and b > 0, we say that a random variable has a beta distribution with parameters a and b if its PDF is

It can be shown that a beta (a,b) random variable, U, satisfies:

Let ? > 0, ? > 0 and ? > 0. Suppose X and Y have joint PDF

(a) Obtain the (marginal) PDF of X in terms of c.

(b) Deduce the value of c.

(c) Identify the conditional distribution of Y given X = x.

(d) Obtain E[XY ] in two different ways.

Solution