Skellam Poisson Difference

Worked Example

Let X and Y be independent Poisson random variables with means a = 2 and b = 1, respectively. We wish to calculate the probability of their difference D = X - Y taking the value k = 1.

The Skellam distribution models this difference. The probability mass function is given by:

P(D = k) = exp(-(a + b)) * sum_(m ≥ max(0, -k)) [ a^(m + k) * b^m / ((m + k)! * m!) ]

For k = 1, the sum starts at m = max(0, -1) = 0. Let's compute the first few terms:

1. For m = 0: a^1 * b^0 / (1! * 0!) = 2^1 * 1^0 / 1 = 2
2. For m = 1: a^2 * b^1 / (2! * 1!) = 2^2 * 1^1 / 2 = 2
3. For m = 2: a^3 * b^2 / (3! * 2!) = 2^3 * 1^2 / 12 = 8 / 12 ≈ 0.6667
4. For m = 3: a^4 * b^3 / (4! * 3!) = 2^4 * 1^3 / 144 = 16 / 144 ≈ 0.1111

Continuing this process and multiplying the sum by exp(-(2 + 1)) = exp(-3) ≈ 0.049787, the total probability converges to approximately 0.23846.

The mean is a - b = 2 - 1 = 1.

The variance is a + b = 2 + 1 = 3.

Experiment

Use the Skellam Poisson Difference interactive tool to explore the distribution of differences.

1. Set Poisson mean 1 to 2 and Poisson mean 2 to 1. Observe that the mean difference is positive and the distribution is slightly skewed.
2. Reverse the means (Poisson mean 1 = 1, Poisson mean 2 = 2). The mean difference becomes negative, and the distribution mirrors the previous state.
3. Set one of the means to 0 (e.g., Poisson mean 1 = 2, Poisson mean 2 = 0). The difference distribution becomes identical to a standard Poisson distribution since one of the random variables is always 0.

Limitations

The Skellam distribution relies on the assumption that the two event counts, X and Y, are independent. In many real-world scenarios, counts may be correlated (e.g., goals scored by two teams in a match, or arrivals in interconnected queuing systems). If the counts are correlated, a bivariate Poisson model is required, and the simple Skellam formula does not apply. Furthermore, calculating the exact probability involves an infinite sum (often represented via modified Bessel functions of the first kind), which must be truncated for practical computation when the means are small, or approximated using normal distributions for very large means.

References

For further reading and the source reference of the Skellam distribution functions used in scientific computations, consider reviewing the documentation provided below.

• SciPy Documentation: scipy.stats.skellam