Skellam Distribution Calculator

Enter the two Poisson means and the value k.

How to use

  1. Enter your values in the fields above.
  2. Press Calculate to see your result instantly.
  3. Use the Share button to copy a link to your result.

About this calculator

The Skellam distribution is the probability distribution of the difference between two independent Poisson random variables, K = X1 − X2. Its probability mass function is P(K=k) = e^(-(μ1+μ2))·(μ1/μ2)^(k/2)·I_k(2√(μ1·μ2)), where I_k is the modified Bessel function of the first kind; its mean is μ1−μ2 and its variance is μ1+μ2.

It's best known for modeling the goal-difference distribution in sports, where each side's score is treated as an independent Poisson process — a standard technique in football (soccer) score-prediction modeling. This calculator is used by sports analytics modelers, statisticians, and researchers comparing the difference between two independent count processes.

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