For example, I am trying to solve the following problem:

If X1...Xn are iid random variables, then what values for "a" and "b" solve the following equation:

E[(X1+...+Xn)] = a*E[X1^2] + b*(E[X1])^2

While we're at it; my car insurance starts at 1000 dollars and reduces by 10% every year that I don't get into an accident. If there is only a 0.05 probability of me getting into an accident any year , does anyone know what the PMF for my total payments up to the year that I get into an accident will be ?

If anyone can help me, then I owe you one.