Compound Distribution of Poisson Distributed Bernoulli Trials has Poisson Distribution

Theorem

Let $N$ be a discrete random variable with a Poisson distribution with expectation $\lambda$.

Let $X_1, X_2, \ldots, X_N$ be pairwise independent discrete random variables each with a Bernoulli distribution with parameter $P$.

Let $S_N : X_1 + X_2 + \cdots + X_N$ be the resulting compound distribution.

Then $S_N$ has a Poisson distribution with expectation $\lambda p$.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): compound distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): compound distribution