For a Binomial(n, p) distribution, which expression gives the mean?

For a Binomial(n, p) distribution, which expression gives the mean?

• A. np(1-p)
• B. np^2
• C. np
• D. n

Answer: (C)

The mean of a Binomial(n, p) is the expected number of successes in n independent trials with success probability p each. View the total successes as a sum of n independent Bernoulli(p) variables. By linearity of expectation, the expected value of that sum is the sum of the expectations, and a Bernoulli(p) variable has mean p. So the average number of successes is n·p, written as np. np(1-p) is the variance, not the mean. np^2 doesn’t represent the mean in general, and would only match np if p were 1. The number n is just the count of trials, not the average number of successes.

The mean of a Binomial(n, p) is the expected number of successes in n independent trials with success probability p each. View the total successes as a sum of n independent Bernoulli(p) variables. By linearity of expectation, the expected value of that sum is the sum of the expectations, and a Bernoulli(p) variable has mean p. So the average number of successes is n·p, written as np.

np(1-p) is the variance, not the mean. np^2 doesn’t represent the mean in general, and would only match np if p were 1. The number n is just the count of trials, not the average number of successes.