A hand-sized method-of-moments estimate

Question

You watch four gaps between events (in minutes): 4, 6, 5, 5. You model them as exponential with rate \lambda, whose mean is 1/\lambda. Use the method of moments to estimate \lambda.

Accepted Answer

Step 1: the sample average. Add and divide: \bar X = \frac{4 + 6 + 5 + 5}{4} = \frac{20}{4} = 5. Step 2: match the mean. The exponential mean is 1/\lambda, so set 1/\lambda = \bar X = 5: \hat\lambda = \frac{1}{5} = 0.2 	ext{ per minute}. That is the whole method here. With one parameter you match the mean; for the exponential the mean is one over the rate, so the estimate is one over the average gap. Takeaway. Average the data, then take the reciprocal: \hat\lambda = 1/\bar X = 0.2.

A hand-sized method-of-moments estimate

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