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


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

  • $\begingroup$ Note that you can also use math typesetting in Question titles. It would make it easier to read what you're asking about. $\endgroup$
    – Sycorax
    May 18 at 16:00
  • 2
    $\begingroup$ Use the tower property to compute $E[SN] = E[E[SN\mid N]]$ in two simple steps. $\endgroup$
    – whuber
    May 18 at 16:21
  • $\begingroup$ See related question here. $\endgroup$
    – Ben
    May 19 at 7:43


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