I want to find the expectation and variance of a sum of N random variables. I know that by the central limit theorem, given that N is known, the expectation should be $N\mu$ and the variance should be $N\sigma^2$ where $\mu$ and $\sigma^2$are the mean and variance of the original distribution. But how should I extend this to the case where N is a random variable from another distribution, say with $\mu_2$ and $\sigma_2^2$?
[Math] Find the expectation and variance of a sum of N random variables where N is itself a random variable
expectationprobabilityvariance
Best Answer
Assuming $N$ is also independent of the ${X_i}$ (Actually, the weaker assumption that $N$ is a stopping time is sufficient), then the variance is given by
$$\sigma^2 = \sigma_N^2\mu_X^2 + \mu_N\sigma_X^2$$
See (http://www.math.unl.edu/~sdunbar1/ProbabilityTheory/Lessons/Conditionals/RandomSums/randsum.shtml) for a proof.