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Given a Poisson process, the probability of obtaining exactly
trials is given by the limit of a binomial distribution
successes in
12,993 entries
Last updated: Fri Aug 20 2010
(1)
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Viewing the distribution as a function of the expected number of successes
(2)
instead of the sample size
for fixed , equation (2) then becomes
(3)
Letting the sample size
become large, the distribution then approaches
(4)
(5)
(6)
(7)
(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 Mathematica as
PoissonDistribution[mu].
As expected, the Poisson distribution is normalized so that the sum of
probabilities equals 1, since
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Poisson Distribution -- from Wolfram MathWorld
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(9)
The ratio of probabilities is given by
(10)
The Poisson distribution reaches a maximum when
(11)
where is the Euler-Mascheroni constant and
leading to the transcendental equation
is a harmonic number,
(12)
which cannot be solved exactly for .
The moment-generating function of the Poisson distribution is given by
(13)
(14)
(15)
(16)
(17)
(18)
so
(19)
(20)
(Papoulis 1984, p. 554).
The raw moments can also be computed directly by summation, which yields
an unexpected connection with the exponential polynomial
and Stirling
numbers of the second kind,
(21)
known as Dobiński's formula. Therefore,
(22)
(23)
(24)
The central moments can then be computed as
(25)
(26)
(27)
so the mean, variance, skewness, and kurtosis are
(28)
(29)
(30)
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Poisson Distribution -- from Wolfram MathWorld
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(31)
(32)
The characteristic function for the Poisson distribution is
(33)
(Papoulis 1984, pp. 154 and 554), and the cumulant-generating function is
(34)
so
(35)
The mean deviation of the Poisson distribution is given by
(36)
The Poisson distribution can also be expressed in terms of
(37)
the rate of changes, so that
(38)
The moment-generating function of a Poisson distribution in two variables is
given by
(39)
If the independent variables
parameters
,
, ...,
,
, ...,
have Poisson distributions with
, then
(40)
has a Poisson distribution with parameter
(41)
This can be seen since the cumulant-generating function is
(42)
(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
(44)
where
is the number of galaxies in a volume
average density of galaxies, and
,
,
is the
, with
is the ratio of gravitational energy to the kinetic energy of peculiar
motions, Letting
gives
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Poisson Distribution -- from Wolfram MathWorld
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motions, Letting
gives
(45)
which is indeed a Poisson distribution with
gives
. Similarly, letting
.
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.
CITE THIS AS:
Weisstein, Eric W. "Poisson Distribution." From MathWorld--A Wolfram Web Resource.
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