medium Probability

Partial Sums I

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

Question

Suppose that $X_1, \dots, X_n$ are all IID random variables with mean $\mu$ and variance $\sigma^2$. Define $S_j = X_1 + \dots + X_j$, with $S_0 = 0$. If possible, compute $\ev{S_k \mid S_n}$, where $0 \leq k \leq n$. Report the answer when $X_1,\dots, X_n \sim \exponential{1}$ IID, $k = 4, n = 10,$ and $S_{10} = 15$.

Practice focus

This Probability problem is tagged Conditional Expectation, 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.