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来源: marketreflections 于 2011-06-27 13:50:44 [] [博客] [旧帖] [给我悄悄话] 阅读数 : (23092 bytes)




















Exponential Distribution

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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 Mathematica 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, ... (Sloane's 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 functionis

















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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