Exponential Distribution

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

			(1)
			(2)
			(3)

and the probability distribution function is

(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

(5)

The raw moments are given by

(6)

the first few of which are therefore 1, , , , , .... Similarly, the central moments are

			(7)
			(8)

where  is an incomplete gamma function and  is a subfactorial, giving the first few as 1, 0, , , , , ... (OEIS A000166).

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

			(9)
			(10)
			(11)
			(12)

The characteristic function is

			(13)
			(14)

where  is the Heaviside step function and  is the Fourier transform with parameters .

If a generalized exponential probability function is defined by

(15)

for , then the characteristic function is

(16)

The central moments are

(17)

and the raw moments are

			(18)
			(19)

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

			(20)
			(21)
			(22)
			(23)

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.