= Binomial thinning
{title2=$Y\mid N\sim\operatorname{Bin}(N,r)$}
Retain each of $N$ objects independently with <probability> $r$, independently of the nonnegative integer-valued <random variable> $N$. The retained count $Y$ has the displayed conditional <binomial distribution>. The <probability generating function of an independent random sum> gives $G_Y(s)=G_N(1-r+rs)$. If $N$ has finite <variance>, the <law of total expectation> and <law of total variance> give
$$
\mathbb E Y=r\mathbb E N,\qquad\operatorname{Var}(Y)=r(1-r)\mathbb E N+r^2\operatorname{Var}(N).
$$
The first term is the average conditional <variance> of the independent retention decisions; the second is the <variance> in their conditional <expected value>.
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