Method of moments when there is just one unknown

Question

A trading desk counts how many client orders arrive each minute. Over 200 minutes it logs 680 orders in total, and the per-minute counts look like independent draws from a Poisson distribution with unknown rate $\lambda$ .

Use the method of moments to estimate $\lambda$ .

Accepted Answer

The one-parameter recipe. The method of moments (MoM) says: write the model's theoretical mean as a function of the unknown, set it equal to the sample average, and solve. With one unknown you need exactly one equation, and the easiest one to use is the mean. Apply it. For a Poisson distribution the mean is the rate itself: \mathbb{E}[X] = \lambda. The sample average is the total divided by the number of minutes: \bar{X} = \frac{680}{200} = 3.4. Matching the two gives \hat{\lambda} = \bar{X} = 3.4 orders per minute. Why this is honest. By the law of large numbers, the sample average creeps to…

Method of moments when there is just one unknown

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