Which statement identifies the geometric distribution?

• A. X ~ Bin(n,p); mean np; variance np(1-p)
• B. X ~ Geom(p); mean p; variance (1-p)/p^2
• C. X ~ Geom(p); mean 1/p; variance (1-p)/p^2
• D. X ~ N(0,1); z = (X - mu)/sigma

Answer: (C)

The geometric distribution describes the number of independent Bernoulli trials needed to get the first success, with each trial having probability p of success. For this parameterization, the expected value (mean) is 1/p, and the variance is (1-p)/p^2. The statement that X follows Geom(p) with mean 1/p and variance (1-p)/p^2 matches these formulas, so it identifies the geometric distribution.

The other statements describe different distributions or use different ideas about the parameters: a binomial distribution has mean np and variance np(1-p) and models a fixed number of trials, not the count until the first success; a normal distribution is continuous and follows a different form; and the one with mean p in place of 1/p does not align with the standard geometric mean for either common parameterization.