= Neyman Type A distribution
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{title2=$G(s)=\exp(\lambda(e^{\theta(s-1)}-1))$}
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Let $N$ have a <Poisson distribution> with parameter $\lambda$, and let $Y_1,Y_2,\ldots$ 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 $Z\mid N\sim\operatorname{Poisson}(\theta N)$. 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.
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