The binomial sum variance inequality states that the variance of the sum of binomially distributedrandom 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.
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^Hoeffding, W. 1956. On the distribution of the number of successes in independent trials. Annals of Mathematical Statistics (27):713–721.