Which of the following expressions correctly represents the mean of a binomial distribution with parameters $n$ and $p$?

• A. $\frac{n}{p}$
• B. $nq$
• C. $p^{\mathrm{n}}$
• D. $np$

Answer: (D)

Verified step by step guidance

1. Recall that a binomial distribution models the number of successes in \( n \) independent Bernoulli trials, each with success probability \( p \).
2. The mean (or expected value) of a binomial distribution is given by the formula \( \mu = n \times p \), where \( n \) is the number of trials and \( p \) is the probability of success on each trial.
3. Examine each given expression to see which matches the formula for the mean:
4. - \( \frac{n}{p} \) represents division, which does not correspond to the mean of a binomial distribution.
5. - \( n \times q \), where \( q = 1 - p \), represents the expected number of failures, not successes.
6. - \( p^n \) is the probability of all trials being successes, not the mean.
7. Therefore, the correct expression for the mean of a binomial distribution with parameters \( n \) and \( p \) is \( n \times p \).