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Date: 29-3-2021
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Given a Poisson process, the probability of obtaining exactly successes in
trials is given by the limit of a binomial distribution
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(1) |
Viewing the distribution as a function of the expected number of successes
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(2) |
instead of the sample size for fixed
, equation (2) then becomes
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(3) |
Letting the sample size become large, the distribution then approaches
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(4) |
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(5) |
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(6) |
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(7) |
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(8) |
which is known as the Poisson distribution (Papoulis 1984, pp. 101 and 554; Pfeiffer and Schum 1973, p. 200). Note that the sample size has completely dropped out of the probability function, which has the same functional form for all values of
.
The Poisson distribution is implemented in the Wolfram Language as PoissonDistribution[mu].
As expected, the Poisson distribution is normalized so that the sum of probabilities equals 1, since
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(9) |
The ratio of probabilities is given by
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(10) |
The Poisson distribution reaches a maximum when
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(11) |
where is the Euler-Mascheroni constant and
is a harmonic number, leading to the transcendental equation
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(12) |
which cannot be solved exactly for .
The moment-generating function of the Poisson distribution is given by
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(13) |
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(14) |
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(15) |
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(16) |
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(17) |
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(18) |
so
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(19) |
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(20) |
(Papoulis 1984, p. 554).
The raw moments can also be computed directly by summation, which yields an unexpected connection with the Bell polynomial and Stirling numbers of the second kind,
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(21) |
known as Dobiński's formula. Therefore,
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(22) |
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(23) |
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(24) |
The central moments can then be computed as
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(25) |
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(26) |
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(27) |
so the mean, variance, skewness, and kurtosis excess are
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(28) |
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(29) |
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(30) |
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(31) |
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(32) |
The characteristic function for the Poisson distribution is
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(33) |
(Papoulis 1984, pp. 154 and 554), and the cumulant-generating function is
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(34) |
so
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(35) |
The mean deviation of the Poisson distribution is given by
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(36) |
The Poisson distribution can also be expressed in terms of
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(37) |
the rate of changes, so that
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(38) |
The moment-generating function of a Poisson distribution in two variables is given by
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(39) |
If the independent variables ,
, ...,
have Poisson distributions with parameters
,
, ...,
, then
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(40) |
has a Poisson distribution with parameter
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(41) |
This can be seen since the cumulant-generating function is
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(42) |
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(43) |
A generalization of the Poisson distribution has been used by Saslaw (1989) to model the observed clustering of galaxies in the universe. The form of this distribution is given by
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(44) |
where is the number of galaxies in a volume
,
,
is the average density of galaxies, and
, with
is the ratio of gravitational energy to the kinetic energy of peculiar motions, Letting
gives
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(45) |
which is indeed a Poisson distribution with . Similarly, letting
gives
.
REFERENCES:
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 532, 1987.
Grimmett, G. and Stirzaker, D. Probability and Random Processes, 2nd ed. Oxford, England: Oxford University Press, 1992.
Papoulis, A. "Poisson Process and Shot Noise." Ch. 16 in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 554-576, 1984.
Pfeiffer, P. E. and Schum, D. A. Introduction to Applied Probability. New York: Academic Press, 1973.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 209-214, 1992.
Saslaw, W. C. "Some Properties of a Statistical Distribution Function for Galaxy Clustering." Astrophys. J. 341, 588-598, 1989.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 111-112, 1992.
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قلة النوم.. ضريبة ثقيلة على صحتك قد تهدد حياتك
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