Discrete Random Variable Representation
Consider the number of AIDS cases that are detected annually in the states of North and South Carolina in the USA. Let X denote the discrete random variable representing the number of cases in a particular year in North Carolina , and let Y denote the number of AIDS cases in a particular year in South Carolina. Assume X and Y are independent random variables, and they have probability distributions:
P(X = x) = (1−π)πx, x = 0,1,2,…
P(Y =y)=(1−θ)θy, y=0,1,2,…
and where π is a constant with value 0 < π < 1 and similarly 0 < θ < 1.
Derive an expression for P (X = Y ).