Let $(X_n : n \in \Bbb N)$ be a sequence of identically distributed random variables, with mean $ \mu$ and variance $\sigma^2 < \infty$. Set $S_0 = 0$ and $S_n=X_1+X_2+...+X_n$ for $n>0$. Let $N$ be a bounded, non-negative integer-valued random variable that is independent of the sequence $(X_n)$.

I've been asked to show that $\Bbb E(S_N^2 | N=n)=n\sigma^2+n^2\mu^2$, yet whenever I try to do this, I keep getting this conditional expectation to be $n^2\sigma^2+n^2\mu^2$. I've tried working from the definition of conditional expectation to no avail, as well as (what seems not very proper) considering simply $\Bbb E(S_n^2)$, but none seem to lead in the right direction.

All my methods seem to lead to $\sigma^2+\mu^2$ multiplied by some function of $N$ with little to do with the distributions of $X$. Any help with this would be much appreciated.

  • $\begingroup$ The variables are independent? $\endgroup$ – leonbloy Feb 25 at 17:34
  • $\begingroup$ I presume so; it doesn't explicitly say so in the original wording, but I feel it makes no sense if they are not. $\endgroup$ – KB399 Feb 25 at 17:34
  • $\begingroup$ Then you need to make that assumption explicit. $\endgroup$ – leonbloy Feb 25 at 17:35

$$S_n^2 = \sum_{i=1}^n X_i^2 + 2 \sum_{i=1}^n \sum_{j \ne i} X_i X_j$$ Since $E[X_i^2] = \sigma^2 + \mu^2$ and $E[X_i X_j] = \mu^2$ for $i \ne j$, we have $$E[S_n^2] = n (\sigma^2 + \mu^2) + n(n-1) \mu^2 = n \sigma^2 + n^2 \mu^2.$$


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