# Exponential Distribution

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Related to Exponential random variable: gamma random variable, Normal random variable, Uniform random variable

## exponential distribution

[‚ek·spə′nen·chəl dis·trə′byü·shən]
(statistics)
A continuous probability distribution whose density function is given by ƒ(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 λex, 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 ex. The mathematical expectation and variance of x are equal to 1/λ and 1/λ2, respectively.

The exponential distribution is the only continuous probability distribution with the property of absence of aftereffect—that is, for any values x1 and x2, the equation

P (X > x1 + x2) = P (X > x1)P (X > x2)

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

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