---
tag: 028Z8WC5Y9
kind: definition
name: Exponential distribution
msc: ["60E05"]
defines: 0E99F51GVB
uses: [04YG690N5V]
sources:
  - url: https://en.wikipedia.org/wiki/Exponential_distribution
    locator: Probability density function
---

## Definition

Let $\lambda > 0$.
The exponential distribution with rate $\lambda$ is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$f(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if } x \ge 0, \\ 0 & \text{if } x < 0. \end{cases}$$
One writes $X \sim \operatorname{Exp}(\lambda)$.

## Remarks

The cumulative distribution function is $F(x) = 1 - e^{-\lambda x}$ for $x \ge 0$, so $P(X > x) = e^{-\lambda x}$.
The mean is $1/\lambda$, the variance is $1/\lambda^2$ and the median is $\ln 2 / \lambda$. Some authors use the scale $\beta = 1/\lambda$ as the parameter.
The exponential distribution is [[memoryless|0ZAZZ34WQS]]: $P(X > s + t \mid X > s) = P(X > t)$ for $s, t \ge 0$.
If $X_1, \dots, X_n$ are independent with $X_i \sim \operatorname{Exp}(\lambda_i)$, then $\min_i X_i \sim \operatorname{Exp}(\lambda_1 + \dots + \lambda_n)$.

## Explanations

### L1 human

The exponential distribution models the time until something happens when it can happen at any moment with the same intensity.
Examples are the decay of a radioactive atom and the time until the next call to a help desk.