AcademyProbability
Academy
Poisson Distribution
Level 1 - Math II (Physics) topic page in Probability.
Principle
A Poisson distribution models a count of events in a region or time interval when events occur at an average rate and are sufficiently independent. The model has no fixed upper bound on the count.
Notation
\(r\)
average event rate per unit interval
\(s\)
interval length, area, volume, or exposure size
\(\lambda=rs\)
mean count in the interval
\(X\)
event count in the interval
\(k\)
a possible count: 0,1,2,...
\(X\sim\operatorname{Po}(\lambda)\)
X has a Poisson distribution with mean lambda
Method
Step 1: Convert rate and interval length to expected count
If the rate is \(r\) per unit and the interval length is \(s\) units, then \(\lambda=rs\). The units cancel to give a dimensionless expected count.
Step 2: Apply the Poisson PMF
Use \(k\) for the count requested. The support is all non-negative integers.
Rules
Mean count
\[\lambda=rs\]
Poisson model
\[X\sim \operatorname{Po}(\lambda)\]
Poisson PMF
\[P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}\]
Mean
\[E[X]=\lambda\]
Variance
\[\operatorname{Var}(X)=\lambda\]
Examples
Question
A radioactive sample emits particles at an average rate of
\[3\]
per minute. Find the probability of exactly \[2\]
emissions in one minute.Answer
Here
\[\lambda=3\cdot1=3\]
\[P(X=2)=e^{-3}\frac{3^2}{2!}.\]
Checks
- A Poisson count has no fixed upper bound.
- The events must be independent enough for the model to be reasonable.
- \(\lambda\) is a dimensionless count expectation, even when computed from a rate and interval length.
- The possible counts are \(0,1,2,\ldots\).
- Mean and variance are both \(\lambda\) for a Poisson random variable.