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The floor on a Poisson rate

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Daily counts of failed orders are independent $Poisson (λ)$ , and you have $50$ days. At $λ = 4$ , what is the Cramér-Rao lower bound on the variance of an unbiased estimator of $λ$ ? Give a fraction.

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Hint

One Poisson count has Fisher information $1/ λ$ about $λ$ .

Solution

One count has $I (λ) = 1/ λ$ , which is $\frac{1}{4}$ at $λ = 4$ .
The sample holds $50 \times \frac{1}{4} = \frac{25}{2}$ units of information.
The bound is $\frac{1}{\frac{25}{2}} = \frac{2}{25}$ .

Answer: $\frac{2}{25}$

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