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Exponential Distribution


ExponentialDistribution

Given a Poisson distribution with rate of change lambda, the distribution of waiting times between successive changes (with k=0) is

D(x)=P(X<=x)
(1)
=1-P(X>x)
(2)
=1-e^(-lambdax),
(3)

and the probability distribution function is

 P(x)=D^'(x)=lambdae^(-lambdax).
(4)

It is implemented in the Wolfram Language as ExponentialDistribution[lambda].

The exponential distribution is the only continuous memoryless random distribution. It is a continuous analog of the geometric distribution.

This distribution is properly normalized since

 int_0^inftyP(x)dx=lambdaint_0^inftye^(-lambdax)=1.
(5)

The raw moments are given by

 mu_n^'=lambda^(-n)n!,
(6)

the first few of which are therefore 1, 1/lambda, 2/lambda^2, 6/lambda^3, 24/lambda^4, .... Similarly, the central moments are

mu_n=(Gamma(n+1,-1))/(elambda^n)
(7)
=(!n)/(lambda^n),
(8)

where Gamma(a,b) is an incomplete gamma function and !n is a subfactorial, giving the first few as 1, 0, 1/lambda^2, 2/lambda^3, 9/lambda^4, 44/lambda^5, ... (OEIS A000166).

The mean, variance, skewness, and kurtosis excess are therefore

mu=1/lambda
(9)
sigma^2=1/(lambda^2)
(10)
gamma_1=2
(11)
gamma_2=6.
(12)

The characteristic function is

phi(t)=F_x{lambdae^(-lambdax)H(x)}(t)
(13)
=(ilambda)/(t+ilambda),
(14)

where H(x) is the Heaviside step function and F_x[f](t) is the Fourier transform with parameters a=b=1.

If a generalized exponential probability function is defined by

 P_((alpha,beta))(x)=1/betae^(-(x-alpha)/beta),
(15)

for x>=alpha, then the characteristic function is

 phi(t)=(e^(ialphat))/(1-ibetat).
(16)

The central moments are

 mu_n^'=e^(alpha/beta)beta^nGamma(n+1,alpha/beta)
(17)

and the raw moments are

mu_n=(beta^nGamma(n+1,-1))/e
(18)
=!nbeta^n,
(19)

and the mean, variance, skewness, and kurtosis excess are

mu=alpha+beta
(20)
sigma^2=beta^2
(21)
gamma_1=2
(22)
gamma_2=6.
(23)

See also

Extreme Value Distribution, Geometric Distribution, Poisson Distribution

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References

Balakrishnan, N. and Basu, A. P. The Exponential Distribution: Theory, Methods, and Applications. New York: Gordon and Breach, 1996.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 534-535, 1987.Sloane, N. J. A. Sequence A000166/M1937 in "The On-Line Encyclopedia of Integer Sequences."Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 119, 1992.

Referenced on Wolfram|Alpha

Exponential Distribution

Cite this as:

Weisstein, Eric W. "Exponential Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentialDistribution.html

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