In statistics, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the Poisson distribution. The probability mass function is
P ( X = n ) = { e − λ λ n + r ( n + r ) ! ⋅ 1 I ( r , λ ) , n = 0 , 1 , 2 , … if r ≥ 0 e − λ λ n + r ( n + r ) ! ⋅ 1 I ( r + s , λ ) , n = s , s + 1 , s + 2 , … otherwise where λ > 0 and r is a new parameter; the Poisson distribution is recovered at r = 0. Here I ( ⋅ , ⋅ ) is the incomplete gamma function and s is the integral part of r. The motivation given by Staff is that the ratio of successive probabilities in the Poisson distribution (that is P ( X = n ) / P ( X = n − 1 ) ) is given by λ / n for n > 0 and the displaced Poisson generalizes this ratio to λ / ( n + r ) .
