Random sum ($Z={\sum }_{j=1}^N{Y}_j$)

A random sum has a random number $N$ of summands, usually with $N$ a nonnegative integer-valued random variable. The empty sum is zero. The dependence between $N$ and the summands must be stated; independent summands do not by themselves imply independence from $N$.

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 giveThe first term is the average conditional variance of the independent retention decisions; the second is the variance in their conditional expected value.

Use the zero-based geometric distribution, with $\mathbb{P}(N=k)=p(1-p{)}^k$ for $k\ge 0$ and $0<p<1$. After binomial thinning with retention probability $0\le r\le 1$, the retained count is again zero-based geometrically distributed, with the displayed success probability. Indeed, writing $q=1-p$, its probability generating function isFor $r=0$ this includes the degenerate success parameter $p^{\prime }=1$ and retained count zero.

For independent identically distributed summands, independent also of the count $N$, conditioning on $N=n$ gives $\mathbb{E}[e^{tZ}\mid N=n]=M_Y(t{)}^n$. The law of total expectation gives the composition of the count's probability generating function with the summand's moment-generating function. This identity is valid as an extended nonnegative expectation; finiteness near zero requires additional exponential-integrability assumptions.