Sum of iid normal variables with index following Poisson distribution

Question

$X_1,X_2,…,X_N,... $ are a sequence of independent normal random variables with mean $\mu$ and variance $\sigma^2$ and index N follows Poisson distribution with parameter $\lambda$. Now, let $S_N = \sum_{i=1}^{N}X_i$. Then what is the mean and variance of $S_N$

I know the sum of iid normal random variables still follows normal distribution. Since index follows Poisson distribution, I don't know how to start with.

Can anyone help me with this?

Thank you.

Answer 1

Let me start with attending you on two general rules:

$$\mathbb EY=\mathbb E[\mathbb E[Y\mid X]]\text{ for mean }\tag1$$and:$$\mathsf{Var}(Y)=\mathbb E\mathsf{Var}(Y\mid X)+\mathsf{Var}\left(\mathbb E[Y\mid X]\right)\text{ for variance }\tag2$$

For $(2)$ see also here.

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We find:

$$\mathbb E\left[\sum_{i=1}^NX_i\mid N=n\right]=\mathbb E\left[\sum_{i=1}^nX_i\mid N=n\right]=\mathbb E\sum_{i=1}^nX_i=n\mu$$

(here the second equality is valid if that there is also independence between $N$ and the $(X_i)_i$ and I suspect that forgot to mention that)

and conclude that:$$\mathbb E\left[\sum_{i=1}^NX_i\mid N\right]=N\mu$$

Now find: $$\mathbb E\sum_{i=1}^NX_i$$ by applying $(1)$.

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Concerning the variance we find: $$\mathsf{Var}\left[\sum_{i=1}^NX_i\mid N=n\right]=\mathsf{Var}\left[\sum_{i=1}^nX_i\mid N=n\right]=\mathsf{Var}\sum_{i=1}^nX_i=n\sigma^2$$

and conclude that: $$\mathsf{Var}\left[\sum_{i=1}^NX_i\mid N\right]=N\sigma^2$$ Now find: $$\mathsf{Var}\sum_{i=1}^NX_i$$ by applying $(2)$.

Answer 2

You also have to assume that $N$ is independent of the sequence $(X_i)_{i\geqslant 1}$. In order to compute the expectaion of $S_N$, we can decompose accordingly to the value that $N$ takes: $$ S_N=\sum_{\ell\geqslant 1}\mathbf{1}_{N=\ell}S_\ell $$ where $S_\ell=\sum_{i=1}^\ell X_i$. Taking the expectation and using independence between $N$ and $S_\ell$, we get $$ \mathbb E\left[S_N\right]=\sum_{\ell\geqslant 1}\mathbb{P}\left(N=\ell\right)\mathbb E\left[S_\ell\right]. $$ Using the assumption on the $X_i$ gives the value of $\mathbb E\left[S_\ell\right]$. Then use the fact that $N$ as a Poisson distribution in order to rearrange this series.

For the variance, it suffices to determine $\mathbb E\left[S_N^2\right]$. To do so, write $$ S_N^2=\sum_{\ell\geqslant 1}\mathbf{1}_{N=\ell}S_\ell^2 $$ and use a similar reasoning as before.