Given a Poisson Distribution with rate of change , the distribution of waiting times between successive
changes (with ) is
which is normalized since
This is the only Memoryless Random Distribution. Define the Mean
waiting time between successive changes as
The Moment-Generating Function is
The Skewness and Kurtosis are given by
The Mean and Variance can also be computed directly
Use the integral
Now, to find
use the integral
If a generalized exponential probability function is defined by
then the Characteristic Function is
and the Mean, Variance, Skewness, and Kurtosis are
See also Double Exponential Distribution
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.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 119, 1992.
© 1996-9 Eric W. Weisstein