Read More
Date: 21-3-2021
1211
Date: 10-4-2021
2417
Date: 1-4-2021
2795
|
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.
|
|
مخاطر خفية لمكون شائع في مشروبات الطاقة والمكملات الغذائية
|
|
|
|
|
"آبل" تشغّل نظامها الجديد للذكاء الاصطناعي على أجهزتها
|
|
|
|
|
تستخدم لأول مرة... مستشفى الإمام زين العابدين (ع) التابع للعتبة الحسينية يعتمد تقنيات حديثة في تثبيت الكسور المعقدة
|
|
|