Ratio of partial sums

MediumRandom Variables Expected Value~5m

Let $X_1, X_2, \dots, X_{40}$ be i.i.d. positive random variables with $\mathbb{E}[X_1] = 2$ and $\mathbb{E}[1/X_1] = 1$ , and let $S_n = X_1 + \dots + X_n$ . Compute $\mathbb{E}\!\left[\dfrac{S_{20}}{S_{40}}\right]$ .

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By exchangeability every $\mathbb{E}[X_i/S_{40}]$ is equal, and they sum to $1$ .
So each is $1/40$ ; $S_{20}$ is $20$ of them. (The given moments are irrelevant.)

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Ratio of partial sums

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