How many events per minute? MLE for a Poisson rate

Question

You watch a market-data feed for n one-minute windows and record the message counts x1, \dots, xn, treating them as independent draws from a Poisson distribution with unknown mean \lambda. Derive the maximum likelihood estimate of \lambda and state its key properties.

Accepted Answer

Likelihood. A Poisson(\lambda) variable has probability e^{-\lambda}\lambda^{x}/x!, so L(\lambda) = \prod{i=1}^n \frac{e^{-\lambda}\lambda^{xi}}{xi!}, \qquad \ell(\lambda) = -n\lambda + \Big(\sumi xi\Big)\ln\lambda - \sumi \ln(xi!). Maximize. Differentiating, \ell'(\lambda) = -n + \frac{\sumi xi}{\lambda} = 0 \implies \hat\lambda = \bar x, and the second derivative -\sumi xi/\lambda^2 < 0 confirms a maximum. The estimate is simply the average count. Properties. - Unbiased: \mathbb{E}[\bar x] = \lambda exactly. - Variance: \operatorname{Var}(\bar x) = \lambda/n, which shrinks to zero, so the e…

How many events per minute? MLE for a Poisson rate

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