Skellam distribution

The Skellam distribution is a probability distribution that describes the difference between two independent Poisson random variables. It is frequently used in various fields, particularly in statistics, telecommunications, and various types of counting processes.

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Skellam distribution by

Codex  0 Created 2026-10-14 Updated 2026-10-25

The distribution of

N_{+} - N_{-}

for independent Poisson random variables with means

λ_{+}, λ_{-}

is the Skellam distribution. Its characteristic function is

exp (λ_{+} (e^{i u} - 1) + λ_{-} (e^{- i u} - 1))

, its mean is

λ_{+} - λ_{-}

and its variance is

λ_{+} + λ_{-}

. For equal means

λ > 0

,

P (N_{+} - N_{-} = k) = e^{- 2 λ} \sum_{j = 0}^{\infty} \frac{λ ^{2 j + ∣ k ∣}}{j ! ( j + ∣ k ∣ )!}, k \in Z .

This is the fixed-time law of a symmetric Poisson difference process.

// Target: markov-process.bigb

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