I am given that X1, X2, X3,....,Xk are independent negative binomial random variables with parameters r1 and p, r2 and p,...and rk and p, respectively. I am trying to find the mgf, pdf, expected value and variance of X=X1+X2+...+Xk.

I am fine with dealing with a single negative binomial variable, so my question is how to deal with a sum of them. Not sure really where to start.

Thanks.