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Exponential distribution

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

$f(x) = \left\{\begin{matrix} \lambda e^{-\lambda x} &,\; x \ge 0 \\ 0 &,\; x < 0 \end{matrix}\right.$

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

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

$P(X > s + t\; |\; X > t) = P(X > s) \;\; \hbox{for all}\ s, t \ge 0.$

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.

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