medium Probability

Overflow I

A release-checked medium problem for training Exchangeability, Expected Value/LOTUS.

Question

Let $X_1,X_2,\dots$ be IID with mean $\mu$, variance $\sigma^2$, and $\ev{\dfrac{1}{S_k}} = a_k$. Define $S_k = \displaystyle \sum_{i=1}^k X_i$. Compute $\ev{\dfrac{S_m}{S_n}}$ when $m \leq n$. For this question, compute it with $\mu = 3, \sigma^2 = 18, a_k = \frac{2}{3k}, m = 25,$ and $n = 50$.

Practice focus

This Probability problem is tagged Exchangeability, Expected Value/LOTUS. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.