Geometric Distribution

#Math

$X \sim \text{Geometric}(p)$

• A geometric random variable represents the number of trials until a first success where each trial is independent and has a probability $p$ of success
• $p_X(x) = (1-p)^{x-1}p, ~ x \in { 1, 2, 3, \ldots }$
• $F_X(x) = 1 - (1 - p)^x$
• $M_X(t) = \frac{pe^t}{1 - (1 - p)e^t},~t < -\ln(1 - p)$
• $\mathbb{E}[X] = \frac{1}{p}$
• $\text{var}(X) = \frac{1 - p}{p^2}$
• A geometric distribution describes the number of trials until a success where each trial has a probability of $p$ of occurring