Binomial Distribution - Covariance Between Two Binomials

Covariance Between Two Binomials

If two binomially distributed random variables X and Y are observed together, estimating their covariance can be useful. Using the definition of covariance, in the case n = 1 (thus being Bernoulli trials) we have

The first term is non-zero only when both X and Y are one, and μ_X and μ_Y are equal to the two probabilities. Defining p_B as the probability of both happening at the same time, this gives

and for n such trials again due to independence

If X and Y are the same variable, this reduces to the variance formula given above.