The Poisson Distribution

The Poisson distribution is one of the most important distributions in Further Statistics. It models the number of events occurring in a fixed interval of time or space when the events happen independently at a constant average rate. This lesson covers the derivation, properties, and applications of the Poisson distribution.

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When to Use the Poisson Distribution

The Poisson distribution is appropriate when:

Condition	Meaning
Events occur singly	Two events cannot happen at exactly the same instant
Events occur independently	One event does not affect the probability of another
Events occur at a constant average rate	The mean rate $\lambda$λ does not change over the interval
The probability of an event in a small interval is proportional to the length of the interval	$P(\text{event in } \delta t) \approx \lambda \delta t$P(event in δt)≈λδt

Examples of Poisson-distributed variables:

• Number of emails received per hour
• Number of typing errors per page
• Number of cars passing a point per minute
• Number of radioactive decays per second

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The Probability Formula

If $X \sim \text{Po}(\lambda)$X∼Po(λ), then:

$P(X = r) = \frac{e^{-\lambda} \lambda^r}{r!} \quad \text{for } r = 0, 1, 2, \ldots$P(X=r)=r!e−λλr​for r=0,1,2,…

where $\lambda > 0$λ>0 is the parameter (the mean number of events in the given interval).

Verification that probabilities sum to 1:

$\sum_{r=0}^{\infty} \frac{e^{-\lambda} \lambda^r}{r!} = e^{-\lambda} \sum_{r=0}^{\infty} \frac{\lambda^r}{r!} = e^{-\lambda} \cdot e^{\lambda} = 1$∑r=0∞​r!e−λλr​=e−λ∑r=0∞​r!λr​=e−λ⋅eλ=1

using the Taylor series for $e^{\lambda}$eλ.

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Mean and Variance

For $X \sim \text{Po}(\lambda)$X∼Po(λ):

Property	Value
$E(X)$E(X)	$\lambda$λ
$\text{Var}(X)$Var(X)	$\lambda$λ
$\text{SD}(X)$SD(X)	$\sqrt{\lambda}$λ​

A key feature of the Poisson distribution is that the mean equals the variance. This property can be used to test whether data follow a Poisson distribution.

Exam Tip: If a question gives you data and asks whether a Poisson model is appropriate, check whether the sample mean is approximately equal to the sample variance. If they differ substantially, the Poisson model may not be suitable.

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Worked Examples

Example 1: Basic Probability Calculation

Emails arrive at a rate of 4 per hour. Find the probability of receiving exactly 6 emails in a one-hour period.

$X \sim \text{Po}(4)$X∼Po(4)

$P(X = 6) = \frac{e^{-4} \times 4^6}{6!} = \frac{0.01832 \times 4096}{720} = \frac{75.07}{720} = 0.1042$P(X=6)=6!e−4×46​=7200.01832×4096​=72075.07​=0.1042

Example 2: Changing the Interval

If emails arrive at 4 per hour, find the probability of receiving exactly 2 emails in 30 minutes.

In 30 minutes, the mean is $\lambda = 4 \times 0.5 = 2$λ=4×0.5=2.

$X \sim \text{Po}(2)$X∼Po(2)

$P(X = 2) = \frac{e^{-2} \times 2^2}{2!} = \frac{0.1353 \times 4}{2} = 0.2707$P(X=2)=2!e−2×22​=20.1353×4​=0.2707

Example 3: Finding $P(X \geq 1)$P(X≥1)

Accidents occur at a rate of 3 per week. Find the probability of at least one accident in a week.