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Poisson distribution

The number of events in a fixed interval when events occur independently at a constant average rate.

Parameters and support

$\lambda$

expected number of events in the interval - $\lambda > 0$

Support

$x \in \{0, 1, 2, \dots\}$

The formulas

p(x)

$$p(x) = \dfrac{e^{-\lambda}\lambda^{x}}{x!}$$

F(x)

$$F(x) = e^{-\lambda}\displaystyle\sum_{k=0}^{\lfloor x \rfloor} \dfrac{\lambda^{k}}{k!}$$

Mean

$$E[X] = \lambda$$

Variance

$$\operatorname{Var}(X) = \lambda$$

MGF

$$M(t) = e^{\lambda\left(e^{t}-1\right)}$$

Where the moments come from

1. Take Binomial($n$, $\lambda/n$) and let $n \to \infty$: $\binom{n}{x}(\lambda/n)^x(1-\lambda/n)^{n-x} \to \frac{e^{-\lambda}\lambda^x}{x!}$.
2. Mean: $E[X] = \lim n \cdot \frac{\lambda}{n} = \lambda$.
3. Variance: $\lim n\frac{\lambda}{n}(1-\frac{\lambda}{n}) = \lambda$, since the correction vanishes.
4. Directly from the MGF: $M(t) = e^{\lambda(e^t-1)}$, so $M'(0) = \lambda$ and $M''(0) = \lambda + \lambda^2$, giving $\operatorname{Var} = \lambda$.
5. Independent Poissons ADD: if $X \sim \text{Poisson}(\lambda_1)$ and $Y \sim \text{Poisson}(\lambda_2)$ are independent, $X + Y \sim \text{Poisson}(\lambda_1+\lambda_2)$.

Worked example

The mean, variance, CDF and moment generating function above are re-derived numerically from this distribution’s own density on every test run - summed over the support for a discrete distribution, integrated by quadrature for a continuous one - and compared with the closed forms printed here. A typo on this page fails that test run.

Traps

• Leaving λ at the stated per-unit rate when the question asks about a different-length interval.
• Adding Poisson parameters for DEPENDENT streams - the addition rule needs independence.
• Confusing the count (Poisson) with the waiting time between events (exponential).

Related

• Exponential distribution
• Gamma distribution
• Binomial distribution

Drill this: the Exam P question bank has original questions on this distribution, and the recall trainer builds its prompts from exactly the formulas above.