A Useful J-Binomial Type Distribution for Non-homogeneous Dichotomous Events

We derive and characterize a new mixed-risk distribution of the total number of Bernoulli outcomes (e.g, failures) for non-homogenous dichotomous events sampled from J non-identical binomial populations with different rates. The resultant distribution can be significantly non-binomial and non-normal, with corresponding consequences on probability calculations, control chart performance, and so on. This random variable mathematically always is under-dispersed compared to its binomial counterpart (with its parameter P equal to a weighted average of the J pi rates). Modified Kullback-Leibler, total absolute deviation, and variance ratio statistics are used to investigate the inaccuracy of binomial and normal approximations on tail probabilities and expected run lengths.