# Random sum of random variables with different variances

If $X_1, X_2,...,X_n$ are independent and have the same expected value, but differing variances, does Wald's equation (that is, $E[X_i]=E[N] E[X]$) still apply?

In particular, my problem to find the distribution of $$\sum_{i=1}^n (-1)^iX_i$$ given the moment generating function for $X_m$.

I don't think that Wald's equation does apply here. I have calculated $X_m$ to be normally distributed with mean $0$ and variance $2m$. Where to go from here?

• I think $N=n$ is fixed is this case and you know explicitly what the distribution of $X_i$ so you don't need Wald's equation here. – Ben May 19 '18 at 2:52

You can write the MGF of the alternating sum in terms of the MGFs for the $X_i$ as follows: $$E(e^{t\sum_{i=1}^n(-1)^i X_i}) = \prod_{i=1}^nE(e^{t(-1)^iX_i})= \prod_{i=1}^nM_{X_i}((-t)^i)$$ where you use independence to factor the expected value.