A probability from a Poisson mean

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Topic: Large-sample estimation · more questions on this topic

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Five independent Poisson observations have sample mean \overline x=1.5. Give the maximum likelihood estimate of \mathbb{P}(X=0)=e^{-\lambda} and the delta-method standard error of that estimate.

Worked answer

The maximum likelihood estimate is e^{-1.5}\approx 0.2231. The delta-method standard error uses h'(\lambda)=-e^{-\lambda} and \operatorname{Var}(\overline X)\approx\lambda/n:

e^{-1.5}\sqrt{1.5/5}\approx 0.2231\times\sqrt{0.3}\approx 0.1222.