Poisson Distribution

Compact study note.

Summary

The Poisson distribution models counts of events in a fixed interval when events occur independently at one constant average rate.^[1]

Prerequisites

• Discrete Random Variable

Definition

$X\sim \mathrm{Poisson}(\lambda )$ with rate parameter $\lambda$ equal to the expected count in the interval.

Notation and Assumptions

The interval is fixed, the average rate is constant, and events do not cluster beyond the model's independence assumptions.

Parameters

$\lambda >0$ .

Support

$\{0,1,2,\dots \}$ .

PMF or PDF

$P(X=k)=e^{-\lambda }{\lambda }^k/k!$ for $k=0,1,2,\dots$ .

CDF

$F_X(k)=\sum _{j=0}^{\lfloor k\rfloor }{e}^{-\lambda }{\lambda }^j/j!$ .

Moments

Moments and MGF:

Essential Result

Poisson counts connect to exponential waiting times in a Poisson process.

Small Example

If defects average $4$ per hour, $P(X=3)=e^{-4}{4}^3/3!\approx 0.1954$ .

Common Mistakes

• Using Poisson when the rate changes across the interval.
• Forgetting that the variance equals the mean under the basic model.

Connections

• Exponential Distribution
• Binomial Distribution
• Distributions

References