Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
A geometric distribution models the number of trials needed to get the first success in a series of independent and identically distributed Bernoulli trials. The probability of success remains constant across all trials.
5 Must Know Facts For Your Next Test
The probability mass function (PMF) is given by $P(X = k) = (1-p)^{k-1} p$ where $p$ is the probability of success, and $k$ is the trial number on which the first success occurs.
The mean (expected value) of a geometrically distributed random variable is $E(X) = \frac{1}{p}$.
The variance of a geometrically distributed random variable is $Var(X) = \frac{1-p}{p^2}$.
Geometric distributions assume that each trial is independent and has only two possible outcomes: success or failure.
The cumulative distribution function (CDF) for a geometric distribution can be expressed as $P(X \leq k) = 1 - (1-p)^k$.
A probability distribution used to describe the number of events occurring within a fixed interval, with events happening independently at a constant rate.