A hand-sized geometric probability estimate

Question

In five rounds it took you 2, 4, 3, 5, 6 tries to get a success. You model the number of tries as geometric with success probability p, whose mean is 1/p. Use the method of moments to estimate p.

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Step 1: the sample average. \bar X = \frac{2 + 4 + 3 + 5 + 6}{5} = \frac{20}{5} = 4. Step 2: match the mean. The geometric mean is 1/p, so set 1/p = \bar X = 4: \hat p = \frac{1}{4} = 0.25. Reading it. It took four tries on average, so about one try in four succeeds. Takeaway. Average the number of tries, then take the reciprocal: \hat p = 1/\bar X = 0.25.

A hand-sized geometric probability estimate

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