Expected ratio of partial sums S40 over S20

Question

Take i.i.d. random variables $X_1,\dots,X_{40}$ with mean $\mathbb{E}[X_1]=2$ , and set $S_n=X_1+\cdots+X_n$ . You are told that $\mathbb{E}[1/S_{20}]=\tfrac{1}{10}$ . Find $\mathbb{E}[S_{40}/S_{20}]$ .

Short answer

$5$ . Write $S_{40}/S_{20}=1+\sum_{i=21}^{40}X_i/S_{20}$ . Each $X_i$ with $i>20$ is independent of $S_{20}$ , so each term has expectation $2\cdot\tfrac{1}{10}$ , giving $1+20\cdot0.2=5$ .

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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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