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The Poisson distribution arises when counting events that occur independently and randomly at a constant rate over time or space. The number of events occuring during a fixed period or in a fixed region is a Poisson(λ) random variable, where λ is the expected number of events. Because Poisson random variables count the number of events occuring, they can only take integer values in the range 0, 1, 2, ...

The function Γ(s, x) in the cdf denotes the upper incomplete gamma function, while the function ⌊x⌋ denotes the "floor" or greatest integer function

Parameter Range Description
λ λ ≥ 0 Rate parameter

Probability Mass Function

P ( X = x ; λ ) = e − λ λ x x !
F ( x ; λ ) = Γ ( ⌊ x + 1 ⌋ , λ ) ⌊ x ⌋ !

Support

x = 0 , 1 , …

Mean

Variance

Example λ
On average, a lottery jackpot is won every four weeks. Let X be the number of jackpot winners in a given week. 0.2500
On average, an accident occurs at a traffic intersection every six months. Let X be the number of accidents in one year. 2.000
On average, 28 people are killed by lightning in the US each year. Let X be the number of people killed in a year. 28.00

X ∼ Poisson(λ)

λ =
Chart of the Poisson distribution Chart area for displaying the Poisson pdf, cdf, and simulation

E(X) = , Var(X) =

A particular feature of the Poisson distribution is that the variance equals the mean. Sample counts therefore vary more as the mean increases.

A particular feature of the Poisson distribution is that the variance equals the mean. Sample counts therefore vary more as the mean increases.

The illustration above shows events occuring randomly in time over a fixed period independently of other events. The expected number of events over this period is given by λ. The actual number of events occurring is a random variable X, which has a Poisson(λ) distribution.

The simulation above shows events occuring randomly in time over a fixed period independently of other events, where λ is the expected number of events over the whole period. The random variable X is the actual number of events occurring during this period, which has a Poisson(λ) distribution. The histogram accumulates the results of each simulation.

Y = limλ→∞ X ∼ Normal(λ, λ) ∑ Xᵢ ∼ Poisson(Σλᵢ)

Chart of the related distribution Chart area for displaying the related pdf, cdf, and simulation

E(Y) = , Var(Y) =

Note that this is expected by the Central Limit Theorem. A Poisson(λ) random variable is the sum of λ independent Poisson(1) random variables, which each have mean 1 and variance 1. By the Central Limit Theorem, the sum of λ i.i.d random variables with finite variance converges to a normal random variable as λ → ∞. So, the sum converges to a normal(λ, λ) random variable as λ → ∞.

The Poisson distribution is shown in red superimposed over the normal limit.

This can be thought of as counting the total number of events occurring in a fixed period for n independent processes with rates λ₁, ..., λₙ. Each process has an expected number of events λi, and the total expected number of events is then λ₁ + ... + λₙ.

The graph above shows the case where λ₁ = λ₂ = ··· = λₙ.