The CDF of the Poisson Distribution

Computing cumulative probabilities for a Poisson random variable: evaluating the CDF directly, using the complement rule for tail probabilities, and finding probabilities over a range of values via differences of CDF values.

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The Poisson CDF

The cumulative distribution function (CDF) of a Poisson random variable $X$ with parameter $\lambda$ gives the probability that $X$ is less than or equal to some non-negative integer $k$ :

F(k) = P(X \leq k) = \sum\limits_{i=0}^{k} \dfrac{e^{-\lambda} \lambda^i}{i!}.

This is just a running sum of the PMF values from $i = 0$ through $i = k$ .

For example, if $X \sim \text{Poisson}(\lambda = 1)$ , then

\begin{align*} F(1) &= P(X = 0) + P(X = 1) \\[3pt] &= \dfrac{e^{-1} \cdot 1^0}{0!} + \dfrac{e^{-1} \cdot 1^1}{1!} \\[3pt] &= e^{-1} + e^{-1} \\[3pt] &= 2e^{-1} \\[3pt] &\approx 0.7358. \end{align*}