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Let $X_1,X_2,...$ be a sequence of independent random variables with $P(X_n = 3^n) = P(X_n = -3^n) = \frac{1}{2}$. Let $S_n = X_1 + ... + X_n$.

  • Compute $E(X_n)$ for each $n$.

My guess for this one is that $E(X_n) = \frac{1}{2}\{\infty - \infty\}$ but that's only because $3^n$ would diverge to $\infty$ and $-3^n$ would diverge to $-\infty$. Is this assumption incorrect or could you tell what I may be missing?

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There is absolutely no problem in computing $E(X_n)$. By the usual method, this is $0$ for all $n$. No infinities involved.

Each $S_n$ also has mean $0$. (You did not ask about the $S_n$.)

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    Ah ok that makes a lot more sense. Thanks! I appreciate it.2012-11-27