Multinomial Distribution of Cell Counts for Independent Identically Distributed Points

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathbb{N}$ be the set of natural numbers with $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, and let $n,m\in\mathbb{N}$. Let $V_1,\dots,V_n$ be independent random variables, each with the same distribution $\nu$. Let $A_1,\dots,A_m$ be pairwise disjoint Borel sets whose union is $\mathbb{R}$, the set of real numbers, and set $p_j=\nu(A_j)$ for $1\le j\le m$.

For $1\le j\le m$ define the cell count

$$S_j=\sum_{i=1}^{n}\mathbf{1}_{\{V_i\in A_j\}},$$

where $\mathbf{1}_{E}$ denotes the function equal to $1$ on the event $E$ and $0$ on its complement. Each $S_j$ is a random variable, as shown in the proof.

Then for every $(n_1,\dots,n_m)\in\mathbb{N}_0^{m}$ with $n_1+\dots+n_m=n$,

$$P\Bigl(\bigcap_{j=1}^{m}\{S_j=n_j\}\Bigr)=\frac{n!}{n_1!\cdots n_m!}\prod_{j=1}^{m}p_j^{\,n_j},$$

with the factorial (convention $0!=1$), the convention $x^{0}=1$, and the finite product notation.