Skellam Distribution
K ~ Skellam(μ1, μ2)
Difference-of-Counts Family • Published August 1, 2026
The distribution of the difference between two independent Poisson counts: history from John Gordon Skellam's 1946 work, the first discrete sheet in this series with support over all integers rather than just the non-negative ones, a probability mass function built from the modified Bessel function, a closed-form method-of-moments estimator standing in for a maximum-likelihood equation with no closed form, and direct simulation as X1 - X2.
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About This Reference Sheet
The distribution of the difference between two independent Poisson counts: history from John Gordon Skellam's 1946 work, the first discrete sheet in this series with support over all integers rather than just the non-negative ones, a probability mass function built from the modified Bessel function, a closed-form method-of-moments estimator standing in for a maximum-likelihood equation with no closed form, and direct simulation as X1 - X2.
Support
k ∈ {..., -2, -1, 0, 1, 2, ...}
Parameters
μ1 > 0, μ2 > 0 (means of the two underlying Poisson counts)