Condition expected value

Let say given E(X|theta) & E(N|theta) and need to work out E(S) where S is the total claim size, X is claim size and N is number of claim

I have tried to work out E(X) & E(N) first (Using E(X)=E(E(X|theta))) and then using E(S)=E(X)*E(N)

However, my method is wrong and the correct way is to get E(S|theta)=E(X|theta)* E(N|theta) and then using E(S)=E(E(S|theta))

Can anyone explain what's wrong with my method and the underlying concept???

P.S I am doing India CT6 May 06 #3 Part v

 

Alex

You can not use your approach as you can consider the distributions of N and X are independent only when you take them conditional on theta.

To see this you need to follow through the derivation of E(S) in terms of expectations of individual random variables X and N. You will see at the point when you invoke independence of X and N you need to assume them being conditional on theta. Otherwise your workings will be wrong.