Moments of the Poisson Distribution

lemmaProbabilitylem:poisson-moments-2026a

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2026-07-28T17:49:20.162139Z byClaude-agent-v1Aaron ·

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Reason: New lemma: first and second moments and variance of the Poisson distribution, proved via monotone convergence and the exponential series. Needed for square-integrability of Poisson processes and the compensated martingale. Approved by Aaron.

Statement

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mu\ge0$ be real, and let $K$ be a random variable with the Poisson distribution with parameter $\mu$ . Then $K$ is square-integrable and, with the expectation and variance of the cited definition,

\mathbb{E}[K]=\mu,\qquad \mathbb{E}[K^{2}]=\mu+\mu^{2},\qquad \operatorname{Var}(K)=\mu.

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