Exponential Distribution
المؤلف:
Balakrishnan, N. and Basu, A. P
المصدر:
The Exponential Distribution: Theory, Methods, and Applications. New York: Gordon and Breach, 1996.
الجزء والصفحة:
...
5-4-2021
1708
Exponential Distribution
Given a Poisson distribution with rate of change 
, the distribution of waiting times between successive changes (with 
) is
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
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
The characteristic function is
where 
is the Heaviside step function and ](https://mathworld.wolfram.com/images/equations/ExponentialDistribution/Inline47.gif)
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
and the mean, variance, skewness, and kurtosis excess are
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.
الاكثر قراءة في الاحتمالات و الاحصاء
اخر الاخبار
اخبار العتبة العباسية المقدسة
