Consider the sum, Z, of two independent binomial random variables, X ~ B and Y ~ B, where Z = X + Y. Then, the variance of Z is less than or equal to its variance under the assumption that p0 = p1, that is, if Z had a binomial distribution. Symbolically,. We wish to prove that We will prove this inequality by finding an expression for Var and substituting it on the left-hand side, then showing that the inequality always holds. If Z has a binomial distribution with parameters n and p, then the expected value of Z is given by Binomial distribution#Mean and variance| E = np and the variance of Z is given by Binomial distribution#Mean and variance| Var = np. Letting n = m0 + m1 and substituting E for np gives The random variables X and Y are independent, so the variance of the sum is equal to the sum of the variances, that is In order to prove the theorem, it is therefore sufficient to prove that Substituting E + E for E gives Multiplying out the brackets and subtracting E + E from both sides yields Multiplying out the brackets yields Subtracting E and E from both sides and reversing the inequality gives Expanding the right-hand side gives Multiplying by yields Deducting the right-hand side gives the relation or equivalently The square of a real number is always greater than or equal to zero, so this is true for all independent binomial distributions that X and Y could take. This is sufficient to prove the theorem. Although this proof was developed for the sum of two variables, it is easily generalized to greater than two. Additionally, if the individual success probabilities are known, then the variance is known to take the form where. This expression also implies that the variance is always less than that of the binomial distribution with, because the standard expression for the variance is decreased by ns2, a positive number.
Applications
The inequality can be useful in the context of multiple testing, where many statistical hypothesis tests are conducted within a particular study. Each test can be treated as a Bernoulli variable with a success probability p. Consider the total number of positive tests as a random variable denoted by S. This quantity is important in the estimation of false discovery rates, which quantify uncertainty in the test results. If the null hypothesis is true for some tests and the alternative hypothesis is true for other tests, then success probabilities are likely to differ between these two groups. However, the variance inequality theorem states that if the tests are independent, the variance of S will be no greater than it would be under a binomial distribution.
