Let X and Y be independent gamma random variables with param

Let X and Y be independent gamma random variables with parameters (alpha_1, lambda) and (alpha_2, lambda), respectively. Define W= X + Y and U = X - Y. Find the joint probability density function of W and U. Show that W is a gamma random variable. Show the U is a beta random variable.

Solution

X and Y are gamma

a) f(x,y) = f(x) f(y) since they are independent

=

b) Consider the characteristic funciton of sum of x and y is the product of char functions.

Hence in exponent it adds up x and y

So sum is a gamma distribution with parameter alpha1, alpha2

Then x+y is also a gamma variable

with parameters (alpha+beta, lemda)