can satisfy. The moment-generating function is given by: which yields the raw moments mk . Define: Then the raw moments mk are The central moments M k are The mean, variance, skewness, and kurtosis excess are respectively: The cumulant-generating function is given by: which yields the cumulants: For the special case when μ1 = μ2, an asymptotic expansion of the modified Bessel function of the first kind yields for large μ: (Abramowitz & Stegun 1972, p. 377). Also, for this special case, when k is also large, and of order of the square root of 2μ, the distribution tends to a normal distribution: These special results can easily be extended to the more general case of different means. Bounds on weight above zero If , with , then Details can be found in Poisson distribution#Poisson races References Irwin, J. O. (1937) "The frequency distribution of the difference between two independent variates following the same Poisson distribution." Journal of the Royal Statistical Society: Series A, 100 (3), 415–416. Karlis, D. and Ntzoufras, I. (2003) "Analysis of sports data using bivariate Poisson models". Journal of the Royal Statistical Society, Series D, 52 (3), 381–393. Karlis D. and Ntzoufras I. (2006). Bayesian analysis of the differences of count data. Statistics in Medicine, 25, 1885–1905. Skellam, J. G. (1946) "The frequency distribution of the difference between two Poisson variates belonging to different populations". Journal of the Royal Statistical Society, Series A, 109 (3), 296. See also Ratio distribution for (truncated) Poisson distributions Discrete distributions Poisson distribution Infinitely divisible probability distributions