# Exponential Distribution

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## exponential distribution

[‚ek·spə′nen·chəl dis·trə′byü·shən]*x*) =

*ae*

^{-ax,}where

*a*> 0 for

*x*> 0, and ƒ(

*x*) = 0 for

*x*≤ 0; the mean and standard deviation are both 1/

*a*.

## Exponential Distribution

a probability distribution on the real line. When *x* ≥ 0, the distribution’s probability density *p(x*) is equal to the exponential function λ*e*^{-λx}, where λ > 0 (hence the name of the distribution). When *x* < 0, *p(x*) = 0. The probability that a random quantity *x* having an exponential distribution will assume values exceeding some arbitrary number *x* is equal to *e*^{-λx}. The mathematical expectation and variance of *x* are equal to 1/λ and 1/λ^{2}, respectively.

The exponential distribution is the only continuous probability distribution with the property of absence of aftereffect—that is, for any values *x*_{1} and *x*_{2}, the equation

*P* (*X* > *x*_{1} + *x*_{2}) = *P* (*X* > *x*_{1})*P* (*X* > *x*_{2})

is satisfied. This characteristic property largely explains, for example, the role that the exponential distribution plays in problems of queuing theory, where the assumption of the exponential distribution of service time is natural. The exponential distribution is closely associated with the concept of a Poisson process. The intervals between successive events in such a process are independent random quantities that have an exponential distribution; here λ is equal to the average number of events per unit time.

### REFERENCE

Feller, W.*Vvedenie v teoriiu veroiatnostei i eeprilozheniia*, 2nd ed., vols. 1-2. Moscow, 1967. (Translated from English.)

A. V. PROKHOROV