The splitting property of the Poisson distribution asserts that if $N \sim \operatorname{Po}(\lambda)$ events are independently classified into $k$ categories with probabilities $p_1, \ldots, p_k$, then the category counts $N_1, \ldots, N_k$ are independent with $N_j \sim \operatorname{Po}(\lambda p_j)$. This is the Poisson counterpart of the multinomial theorem for binomial counts, but with the remarkable additional feature that the category counts are independent — a property that is emphatically NOT shared by the multinomial distribution (where the counts must sum to the fixed total $N$). The proof computes the joint probability generating function of $(N_1, \ldots, N_k)$ by conditioning on $N$: $\mathbb{E}[\prod_j z_j^{N_j}] = \mathbb{E}[\mathbb{E}[\prod_j z_j^{N_j} \mid N]] = \mathbb{E}[(\sum_j p_j z_j)^N] = \exp(\lambda(\sum_j p_j z_j - 1)) = \prod_j \exp(\lambda p_j(z_j - 1))$, which factors as the product of $\operatorname{Po}(\lambda p_j)$ generating functions, confirming independence. The splitting property is the key structural result for [Poisson Random Measures](/theorems/1194). It explains why restricting a Poisson random measure to a subset produces an independent Poisson random measure: points in the subset and its complement are independent Poisson counts with parameters proportional to the measure of each part. This property makes the Poisson random measure the canonical model for independent scattering and is the basis for the theory of marked point processes, thinning operations, and the decomposition of compound Poisson processes into independent components.