Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, let $\mathbb{N}$ be the set of \reftext{def:natural-numbers-2026a}{natural numbers} with $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, and let $r\in\mathbb{N}$. Let $X_1,\dots,X_r$ be random variables such that $X_i(\omega)\in\mathbb{N}_0$ for every $\omega\in\Omega$ and every $i$. Suppose that for each $i$ there is a function $g_i:\mathbb{N}_0\to[0,1]$ with $\sum_{c=0}^{\infty}g_i(c)=1$ such that for all $(c_1,\dots,c_r)\in\mathbb{N}_0^{r}$,

with the \reftext{def:finite-product-notation-2026a}{finite product notation}. Then:

\textbf{1.} for every $i$ and every \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel set} $B$,

the sum over the countable index set $B\cap\mathbb{N}_0$ being the supremum of its finite partial sums (all terms are nonnegative); in particular $P(X_i=c)=g_i(c)$;

\textbf{2.} the random variables $X_1,\dots,X_r$ are \reftext{def:independence-events-rvs-2026a}{independent}.