Exponential distribution

The story

If events arrive as a Poisson process with rate $\lambda$ per unit time, the gap until the next one is Exponential. Its density $\lambda e^{-\lambda x}$ is highest at zero and decays steadily, so short waits are common and long waits are rare but always possible. The mean wait is $1/\lambda$ and the standard deviation is also $1/\lambda$ .

Its signature property is memorylessness: $P(X > s + t \mid X > s) = P(X > t)$ . If a bulb with an Exponential lifetime has already lasted 1000 hours, its remaining life has the same distribution as a brand-new bulb's. It is the only continuous distribution with this property.

Formulas

Support

x \ge 0

PDF

f(x) = \lambda e^{-\lambda x}

CDF

F(x) = 1 - e^{-\lambda x}

Expectation

\mathbb{E}[X] = \frac{1}{\lambda}

Variance

\operatorname{Var}(X) = \frac{1}{\lambda^2}

Mode

0

Parameters

Parameter	What it controls
$\lambda$	The rate: the average number of events per unit time. A higher rate means shorter waits, with mean $1/\lambda$ and variance $1/\lambda^2$ .

Where it shows up

Time between customers arriving at a shop counter.
Time until the next radioactive decay in a sample.
The lifetime of an electronic component that does not wear out.
Time between requests hitting a web server.

Try this

Shade from 0 to the mean $1/\lambda$ . Whatever $\lambda$ you choose, the probability is $1 - e^{-1} \approx 0.632$ , so most waits are shorter than average: the long tail drags the mean upwards.
Set $\lambda = 0.5$ and press Draw 1000. Check that the sample mean is close to 2 and the sample variance close to 4, the square of the mean.
Switch to Sample means with $n = 2$ , then $n = 30$ . The averages stop piling up at zero and form a bell around $1/\lambda$ : the Central Limit Theorem straightens out even a strongly skewed shape.

Related distributions

The Exponential with rate $\lambda$ is the Gamma with $k = 1$ and $\theta = 1/\lambda$ .
It gives the gaps between events in a Poisson process, whose counts in a window of length $t$ are $\text{Poisson}(\lambda t)$ .
Chi-squared with $k = 2$ is the Exponential with rate $1/2$ (mean 2).
It is the continuous analogue of the Geometric: both are memoryless waiting times.