Question

Various proof of inequalities of expectation of binomial distribution

Answer 1

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 pparameters. 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.The lack of uniformity in success probabilities across independent trials leads to a smaller variance.and is a special case of a more general theorem involving the expected value of convex functions. n 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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