A geometric random variable consists of independent Bernoulli trials, each with the same probability of success p, repeated until the first success is obtained *inclusive* of the success. For the distribution exclusive of the first success, see the [[geometric 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 $X \sim Geom(p)$ > [!warning] > Note that this notation is equivalent to the notation of the geometric distribution. The pmf and expectation will differ depending on whether you count the first success or not. ## PMF $P(X=x) = p(1-p)^{x-1}$ ## Expectation $E(X) = \frac{1}{p}$ ## Variance $V(X) = \frac{(1-p)}{p^2}$