What are the mean and variance of a binomial distribution?

• A. Mean np; Variance np(1-p)
• B. Mean p; Variance p(1-p)
• C. Mean n; Variance p
• D. Mean np; Variance p

Answer: (A)

In a binomial distribution with n trials and probability p of success on each trial, the mean is np and the variance is np(1-p). This comes from viewing the binomial as the sum of n independent Bernoulli(p) variables. Each Bernoulli has mean p and variance p(1-p); adding the n independent trials gives E[X] = np and Var(X) = n p (1-p). The mean must scale with n, and the variance incorporates both n and p through the term p(1-p); options that use p alone or drop n miss these relationships.