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Skellam Distribution Calculator

Enter the two Poisson means and the value k.

Result

0.232379
0.4000
2.6000

How it is calculated

If X1 and X2 are independent Poisson counts with means μ1 and μ2, their difference K = X1 − X2 follows the Skellam distribution. The tool evaluates P(K = k) = e−(μ1+μ2) × (μ1/μ2)k/2 × I|k|(2√(μ1·μ2)), where I is the modified Bessel function of the first kind, summed from a power series in logarithms so that large means still work (for example 100 and 90 give a normal probability). It also reports mean = μ1 − μ2 and variance = μ1 + μ2. Both means must be above zero and not larger than 1,000,000. The value k is rounded to the nearest whole number (1.4 is used as 1), can be negative, meaning the second count is larger, and must not exceed 1,000,000 in size. The probability is given to six decimals, so a very unlikely k shows 0.000000.

Worked example

A football match is modelled with an average of μ1 = 1.5 goals for the home team and μ2 = 1.1 for the away team. What is the chance the home team wins by exactly one goal, k = 1? The calculator gives P(K = 1) = 0.232379, about 23.2%. The mean difference is 1.5 − 1.1 = 0.4000 and the variance is 1.5 + 1.1 = 2.6000. For comparison, k = 0 (a draw) gives 0.257667 and k = −1 (an away win by one) gives 0.170411.

Try this example →

Common mistakes

  • Typing the variance or the standard deviation instead of the means. Both inputs are Poisson means, the average counts of each side.
  • Mixing up the sign of k: positive k means the first count (μ1) is larger, negative k means the second one is.
  • Expecting the probability of "at least" or "more than" a margin. The tool gives only the chance of exactly k; for a win by 1 or more, add the values for k = 1, 2, 3 and so on.
  • Using the Skellam model for dependent counts. It assumes the two counts are independent, so strongly linked teams or events break that assumption.
  • Entering a mean of 0 or a negative number; both means must be above zero, otherwise no result is shown.

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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Comments (1)

  • Editorial team

    The tool returns one margin at a time, but adding margins gives full match odds. With means of 1.5 and 1.1, summing k = 1 upward gives about 0.4642 for a home win, k = 0 gives 0.2577 for a draw and summing k = -1 downward gives about 0.2781 for an away win, which together make 1. Remember the model assumes independent scoring and fixed averages. Are you modelling football or another sport, and where do your means come from?

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