I am busy with ruin theory. S(t) is the aggregate claim size after t years, where Xi is the individual claim size (with mean and variance given) and N(t) is the number of claims that follow a Poisson distribution with parameter lambda, and it is assumed that lambda has an exponential distribution with mean given.

Now since N(t) has a Poisson distribution, S(t) has a Compound Poisson distribution with parameter lambda, right?

Then is the expected value of S(t): E[S(t)] = E[N(t)]*E[Xi] = (E[lambda]*t)*E[Xi] ?

I am spesifically confused about the E[N(t)] part, does it include the t variable even though it is only distributed Poisson(lambda) or not? And then do you use E[lambda] or only lambda in calculating the E[N(t)]?

Also, the variance of N(t), is it equal to the variance of lambda or the expected value of lambda? And again should it be multiplied by t even though t is not given as part of the parameter in the question?

Would be so glad if anyone can help. And please ask if there is any more information needed to answer the question.