A geometric random variable consists of independent Bernoulli trials, each with the same probability of success $p$, repeated until the first success is obtained *exclusive* of the success. For the distribution inclusive of the first success, see the [[first success distribution]]. - Each trial is identical, and can result in a success or failure - The probability of success is constant from one trial to the next - The trials are independent - Trials are repeated until the first success ## Notation $X \sim Geom(p)$ ## PMF $P(X=x) = p(1-p)^{x}$ ## Expectation $E(X) = \frac{1-p}{p}$ ## Variance $V(X) = \frac{(1-p)}{p^2}$