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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)

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

F(x)

$$F(x)=e^{-\lambda }\sum _{k=0}^{\lfloor x\rfloor }\frac{{\lambda }^k}{k!}$$

Mean

$$E[X]=\lambda$$

Variance

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

MGF

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

Where the moments come from

1. Take Binomial($n$, $\lambda /n$) and let $n\to \infty$: $\left(\genfrac{}{}{0ex}{}{n}{x}\right)(\lambda /n{)}^x(1-\lambda /n{)}^{n-x}\to \frac{e^{-\lambda }{\lambda }^x}{x!}$.
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 $M^{\prime }(0)=\lambda$ and $M^{\prime \prime }(0)=\lambda +{\lambda }^2$, giving $\mathrm{Var}=\lambda$.
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 the build.

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.