For a Binomial(n, p), what are its mean and variance?

For a Binomial(n, p), what are its mean and variance?

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

Answer: (A)

The key idea is that a binomial(n, p) counts how many successes occur in n independent trials each with probability p of success. The distribution is the sum of n independent Bernoulli(p) variables. Each Bernoulli has expected value p and variance p(1−p). Since the trials are independent, the expected value of the sum is the sum of the expected values, giving np. Similarly, the variances add, giving n p(1−p). Therefore, the mean is np and the variance is np(1−p). This matches the given correct choice. The other options mix up the scaling by n or replace the binomial with a Bernoulli result.

The key idea is that a binomial(n, p) counts how many successes occur in n independent trials each with probability p of success. The distribution is the sum of n independent Bernoulli(p) variables. Each Bernoulli has expected value p and variance p(1−p). Since the trials are independent, the expected value of the sum is the sum of the expected values, giving np. Similarly, the variances add, giving n p(1−p). Therefore, the mean is np and the variance is np(1−p). This matches the given correct choice. The other options mix up the scaling by n or replace the binomial with a Bernoulli result.