The binomial sum variance inequality states that the variance of the sum of binomially distributed random variables will always be less than or equal to the variance of a binomial variable with the same n and p parameters. In probability theory and statistics, the sum of independent binomial random variables is itself a binomial random variable if all the component variables share the same success probability. If success probabilities differ, the probability distribution of the sum is not binomial.[1] The lack of uniformity in success probabilities across independent trials leads to a smaller variance.[2][3][4][5][6] and is a special case of a more general theorem involving the expected value of convex functions.[7] In some statistical applications, the standard binomial variance estimator can be used even if the component probabilities differ, though with a variance estimate that has an upward bias.
- ^ Butler, Ken; Stephens, Michael (1993). "The distribution of a sum of binomial random variables" (PDF). Technical Report No. 467. Department of Statistics, Stanford University. Archived (PDF) from the original on April 11, 2021.
- ^ Nedelman, J and Wallenius, T., 1986. Bernoulli trials, Poisson trials, surprising variances, and Jensen’s Inequality. The American Statistician, 40(4):286–289.
- ^ Feller, W. 1968. An introduction to probability theory and its applications (Vol. 1, 3rd ed.). New York: John Wiley.
- ^ Johnson, N. L. and Kotz, S. 1969. Discrete distributions. New York: John Wiley
- ^ Kendall, M. and Stuart, A. 1977. The advanced theory of statistics. New York: Macmillan.
- ^ Drezner, Zvi; Farnum, Nicholas (1993). "A generalized binomial distribution". Communications in Statistics - Theory and Methods. 22 (11): 3051–3063. doi:10.1080/03610929308831202. ISSN 0361-0926.
- ^ Hoeffding, W. 1956. On the distribution of the number of successes in independent trials. Annals of Mathematical Statistics (27):713–721.
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