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Distributionshard
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Let X∼Binomial(n,p)X \sim \text{Binomial}(n, p)X∼Binomial(n,p) and Y∼Binomial(m,p)Y \sim \text{Binomial}(m, p)Y∼Binomial(m,p) be independent binomial random variables. Let S=X+YS = X + YS=X+Y. What is the conditional variance Var(X∣S=s)Var(X \mid S = s)Var(X∣S=s)?