In probability theory and statistics, the exponential distribution is a continuous probability distribution with the probability density function
- <math>
where λ > 0 is a parameter of the distribution.
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/λ2.)
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
An important property of the exponential distribution is that it is memoryless. This means that if a random variable X is exponentially distributed, its conditional probability obeys
- <math>P(X > s + t\; |\; X > t) = P(X > s) \;\; \hbox{for all}\ s, t \ge 0. </math>
In other words, the chance that the state change is going to happen in the next s seconds is unaffected by the amount of time that has already elapsed.