Variance of a sum of a random number of iid random variables [duplicate]

Question

I tried to get an expression for the variance of a sum of a random number of iid random variables. My question is whether it is correct and, if not, what is wrong or what additional assumptions might be missing.

Specifically, let:

$$S=\sum_1^N{X_i},$$ where $N$ is a non-negative integer-valued random variable.

Suppose that the distributions of both $N$ and $X$ are known (and $X_i$ are iid), I want to know the value of the variance of $S$.

Is the following solution correct?

$$V(S)=V(X)E(N)+E(X)^2V(N)$$

Answer 1

Use the definition of variance $\sigma^{2} = \frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}$. For an R.V. $X$, we have $\mathbb{V}(X)=\mathbb{E}[(X - \mathbb{E}[X])^{2}]$.

Answer 2

From the Law of total variance we have the decomposition: $$ \text{Var}(S) = \mathbb{E}[\text{Var}(S \mid N)] + \text{Var}[\mathbb{E}(S \mid N)] $$ Given $N$, $S$ is simply a sum of $N$ iid random variables so, $$ \text{Var}(S \mid N) = N \times \text{Var}(X) \\ \mathbb{E}(S \mid N) = N \times \mathbb{E}(X) $$ The two quantities above are random variables but only as functions of $N$. Thus, since $\text{Var(X)}$ and $\mathbb{E}(X)$ are constants, we have:

$$ \mathbb{E}[\text{Var}(S \mid N)] = \mathbb{E}(N) \text{Var}(X) $$ and $$ \text{Var}[\mathbb{E}(S \mid N)]=\mathbb{E}(X)^2 \text{Var}(N) $$ Finally, $$ \text{Var}(S) = \mathbb{E}(N) \text{Var}(X) + \mathbb{E}(X)^2 \text{Var}(N) $$

So yes, your solution is correct.

Here is some piece of R code where $N$ is Poisson distributed and the $X_i$ are $\mathcal{N}(4,1)$:

s<-sapply(1:10000,function(i){
      N<-rpois(1,20)
      S<-sum(rnorm(N,4,1))
      return(S)
   })
var(s) #should be close to 340
20*1 + 16*20