I'm just having trouble with this questions it's more proof then anything:
Using probability generating functions verify the following results for independent random variables, and hence derive the expectation and variance of the sum.
(a) If X1,....,Xk are Bernoulli random variables with common parameter p, then X1 +...+Xk has a binomial distribution with parameters (k,p)
(b) If X1,...,Xk are geometric random variables with common parameter p, then X1+....Xk has a negative binomal distribution with parameters (k,p)
(c) If X1,...,Xk are Poisson random variables with common parameters lambda1,....,lambdak, then X1+...+Xk also has a Poisson distribution with parameter lambda1+...lambdak.
Don't really understand the question so any help would be appreciated.