Abstract
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 Np̄ and N(p̄ q̄ + N(N–1) V (p, where p̄ = 1–q̄ 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 p̄ and N p̄ q̄ – NV′(p), where V′(p) is the variance of p within experiments.
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References
Edwards, A. W. F., Ann. Hum. Genet., 23, 6 (1958).
Parkes, A. S., Biometrika, 15, 373 (1923).
David, F. N., “Probability Theory for Statistical Methods” (Cambridge University Press, 1951).
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EDWARDS, A. The Meaning of Binomial Distribution. Nature 186, 1074 (1960). https://doi.org/10.1038/1861074a0
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DOI: https://doi.org/10.1038/1861074a0
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