What is Var(X) for Binomial(n, p)?

What is Var(X) for Binomial(n, p)?

• A. np
• B. np(1-p^2)
• C. n(1-p)
• D. np(1-p)

Answer: (D)

Think about what X represents and how variance behaves. A binomial variable X ~ Binomial(n, p) counts the number of successes in n independent Bernoulli trials with success probability p. Each Bernoulli(p) trial has variance p(1 − p). Since X is the sum of n independent Bernoulli trials, the variances add: Var(X) = n · p(1 − p) = np(1 − p). This is why this expression is the correct measure of spread for X. The other forms mix up the mean with the variance or come from incorrect manipulations, so they don’t match how variance behaves in a binomial setting. If you needed the standard deviation, it would be sqrt(np(1 − p)).

Think about what X represents and how variance behaves. A binomial variable X ~ Binomial(n, p) counts the number of successes in n independent Bernoulli trials with success probability p. Each Bernoulli(p) trial has variance p(1 − p). Since X is the sum of n independent Bernoulli trials, the variances add: Var(X) = n · p(1 − p) = np(1 − p). This is why this expression is the correct measure of spread for X. The other forms mix up the mean with the variance or come from incorrect manipulations, so they don’t match how variance behaves in a binomial setting. If you needed the standard deviation, it would be sqrt(np(1 − p)).