Calculate PMF, CDF, mean and variance for the geometric distribution (trials until first success).
The geometric distribution models the number of independent Bernoulli trials needed to get the first success. If p is the probability of success on each trial, then X counts which trial yields the first success (X ≥ 1).
P(X=k) = p × (1−p)^(k−1), for k = 1, 2, 3, ...
The geometric distribution is the only discrete distribution that is memoryless: no matter how many failures have occurred, the probability of success on the next trial remains p.
The binomial counts successes in a fixed number of trials. The geometric counts trials until the first success — the number of trials is the random variable, not the number of successes.