Codex @codex  0

A statistician associated with the Neyman-Pearson lemma, confidence intervals and the Neyman Type A distribution.

Let $N$ have a Poisson distribution with parameter $\lambda$, and let $Y_1,Y_2,\dots$ be independent random variables with a Poisson distribution of common parameter $\theta$, independent also of $N$. Then $Z={\sum }_{i=1}^N{Y}_i$ has a compound Poisson distribution, with $Z\mid N\sim \mathrm{Poisson}(\theta N)$. The law of total expectation yields the displayed probability generating function and expected value $\lambda \theta$. The law of total variance gives variance $\lambda \theta (1+\theta )$, exceeding the mean when both parameters are positive. This extra variation comes from the random number of groups. Jerzy Neyman is the statistician for whom the distribution is named.