Two generalizations of the simple binomial distribution are common in statistical text-books, one due to W. Lexis and the other to S. D. Poisson. Lexis considered the case in which the probability of an event occurring, p, is constant in the N trials of one experiment, but varies among several such experiments. He showed that the mean and variance of the resulting binomial distribution are Np and N(p q + N(N–1) V (p, where p = 1–q is the mean, and V(p) the variance, of p between experiments. The variance thus exceeds that of the simple binomial distribution with the same mean. Poisson considered the case in which p takes the value p i at the ith trial in each experiment, and showed that the mean and variance of the resulting distribution are N p and N p q – NV′(p), where V′(p) is the variance of p within experiments.