Expected share of the first half of a sum

Question

Random variables $X_1,\dots,X_{40}$ are i.i.d.; their common mean is $2$ , and the mean of their reciprocal $1/X_i$ is $1$ . Writing $S_n$ for the partial sum $X_1+\dots+X_n$ , find the expected value of the ratio $S_{20}/S_{40}$ .

Short answer

$\tfrac12$ . By exchangeability every ratio $X_i/S_{40}$ has the same mean, and the $40$ ratios add up to $1$ , so each has mean $\tfrac1{40}$ . Summing $20$ of them gives $\tfrac12$ ; the moment information given is a distraction.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Sample Spaces, Events, and the Probability Axioms

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