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Date: 22-2-2021
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Given a Poisson distribution with rate of change , the distribution of waiting times between successive changes (with ) is
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and the probability distribution function is
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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
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the first few of which are therefore 1, , , , , .... Similarly, the central moments are
(7) |
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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
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The characteristic function is
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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
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and the raw moments are
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and the mean, variance, skewness, and kurtosis excess are
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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.
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دراسة يابانية لتقليل مخاطر أمراض المواليد منخفضي الوزن
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اكتشاف أكبر مرجان في العالم قبالة سواحل جزر سليمان
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اتحاد كليات الطب الملكية البريطانية يشيد بالمستوى العلمي لطلبة جامعة العميد وبيئتها التعليمية
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