Poisson Distribution
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)
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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)
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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)
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The ratio of probabilities is given by
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(10)
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The Poisson distribution reaches a maximum when
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(11)
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where is the Euler-Mascheroni constant and
is a harmonic number, leading to the transcendental equation
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(12)
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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)
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so
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(19)
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(20)
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(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)
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known as Dobiński's formula. Therefore,
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(22)
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(23)
|
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(24)
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The central moments can then be computed as
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(25)
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(26)
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(27)
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so the mean, variance, skewness, and kurtosis 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)
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The characteristic function for the Poisson distribution is
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(33)
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(Papoulis 1984, pp. 154 and 554), and the cumulant-generating function is
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(34)
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so
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(35)
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The mean deviation of the Poisson distribution is given by
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(36)
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The Poisson distribution can also be expressed in terms of
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(37)
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the rate of changes, so that
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(38)
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The moment-generating function of a Poisson distribution in two variables is given by
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(39)
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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)
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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
.
SEE ALSO: Binomial Distribution, Erlang Distribution, Poisson Process, Poisson Theorem
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.
Referenced on Wolfram|Alpha: Poisson Distribution
CITE THIS AS:
Weisstein, Eric W. "Poisson Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html
1重 0-1分布
N重 二项分布 , 系数为阶乘降/阶乘增, 从0开始
无限重 v=Np, 泊松分析, 先确定N,再确定对应的p, 再得v, 此时才有泊松分布公式可用