In a binomial distribution, which parameter pair defines its shape and mean?

• A. mu and sigma; mean mu
• B. lambda and p
• C. n and p; mean np
• D. n and p; mean p

Answer: (C)

In a binomial distribution, the shape and the location of the center are determined by two parameters: n, the number of trials, and p, the probability of success on each trial. The mean of the distribution is np, so these two parameters together define both how the distribution looks and where its average lies.

Other options mix parameters from different distributions or misstate the mean: mu and sigma belong to the normal distribution (mean is mu), lambda is the rate parameter for the Poisson distribution, and having the mean as p ignores the effect of counting up to n trials. Therefore, the pair that defines both the shape and the mean is n and p, with mean np.