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 $N$.

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.