In probability theory and statistics, the **exponential distribution** is a continuous probability distribution with the probability density function

The graph below shows the probability density function for λ equal to 0.5, 1.0, and 1.5:

The expected value and standard deviation of an exponential random variable are both 1/λ (and thus its variance is 1/λ^{2}.)

The exponential distribution is used to model Poisson processes, which are situations in which an object initially in state A can change to state B with constant probability per unit time λ. The time at which the state actually changes is described by an exponential random variable with parameter λ. Therefore, the integral from 0 to *T* over *f* is the probability that the object is in state B at time *T*.

The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state.

Examples of variables that are approximately exponentially distributed are:

- the time until you have your next car accident
- the time until you get your next phone call
- the distance between mutations on a DNA strand
- the distance between roadkill

*memoryless*. This means that if a random variable

*X*is exponentially distributed, its conditional probability obeys

*s*seconds is unaffected by the amount of time that has already elapsed.