---
tag: 0XCVV5Q500
kind: definition
name: Geometric distribution
msc: ["60E05"]
defines: 04P2SR252N
uses: [0WF4EJP91W, 0PHC9FE2V5]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_distribution
    locator: Definition
---

## Definition

Let $p \in (0, 1]$.
The geometric distribution with parameter $p$ is the distribution on $\{0, 1, 2, \dots\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(k) = (1 - p)^k \, p \quad \text{for } k = 0, 1, 2, \dots$$

## Remarks

It is the distribution of the number of failures before the first success in independent [[Bernoulli trials|0PHC9FE2V5]] with success probability $p$.
The other common convention counts the trials up to and including the first success. That number $Y = X + 1$ has $P(Y = k) = (1 - p)^{k-1} p$ for $k \ge 1$.
For $X$ as in the definition, $P(X \ge k) = (1 - p)^k$, the mean is $(1 - p)/p$ and the variance is $(1 - p)/p^2$.
The geometric distribution is the only distribution on $\{0, 1, 2, \dots\}$ with the [[memoryless property|0ZAZZ34WQS]].

## Explanations

### L1 human

Toss a coin until the first head. The number of tails before it has the geometric distribution.
Each extra tail multiplies the probability by $1 - p$, so the probabilities form a geometric sequence.