= Geometric-sum moment-generating function
{title2=$M_S(t)=pM_X(t)/(1-(1-p)M_X(t))$}
For a <geometric distribution> count on the positive integers, independent of <independent and identically distributed random variables> $X_i$, the <moment-generating function> of $S=\sum_{i=1}^NX_i$ is the displayed expression. Condition on the count and sum a <geometric series>. For real $t$, the finite domain requires $M_X(t)<\infty$ and $(1-p)M_X(t)<1$. Positive summands always permit $t\leq0$, even when no positive exponential moment exists; this gives a <Laplace transform of a nonnegative random variable>. The aggregate has no mass at zero when all summands are strictly positive.
Back to article page
