# $E(SN)$ for aggregate claim amount $S$, $S=X_{1}+...+X_{n}, X_{i}$ are iid [duplicate]

Consider the following model for aggregate claim amounts $$S$$:

$$S=X_{1}+X_{2}+...+X_{N}$$

where the $$X_{i}$$ are independent, identically distributed random variables representing individual claim amounts and N is a random variable, independent of the $$X_{i}$$ and representing the number of claims. Let $$X$$ have mean $$\mu_{X}$$ and let $$N$$ have mean $$\mu_{N}$$ and variance $$\sigma^2_{N}$$. Then $$E(SN)$$ equals?

I tried to solve it as follows $$E(SN)=E(\sum X_{i}*N)=NE(X_{i})E(N)$$. Is this correct given that N is also a random variable?

The answer is $$\mu_{X}(\mu^2_{N}+\sigma^2_{N})$$

• Note that you can also use math typesetting in Question titles. It would make it easier to read what you're asking about.
– Sycorax
May 18 at 16:00
• Use the tower property to compute $E[SN] = E[E[SN\mid N]]$ in two simple steps.
– whuber
May 18 at 16:21
• See related question here.
– Ben
May 19 at 7:43