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Poisson Distribution

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Given a Poisson process, the probability of obtaining exactly n successes in N trials is given by the limit of a binomial distribution

P_p(n|N)=(N!)/(n!(N-n)!)p^n(1-p)^(N-n).
(1)

Viewing the distribution as a function of the expected number of successes

nu=Np
(2)

instead of the sample size N for fixed p, equation (2) then becomes

P_(nu/N)(n|N)=(N!)/(n!(N-n)!)(nu/N)^n(1-nu/N)^(N-n),
(3)

Letting the sample size N become large, the distribution then approaches

P_nu(n)=lim_(N->infty)P_p(n|N)
(4)
=lim_(N->infty)(N(N-1)...(N-n+1))/(n!)(nu^n)/(N^n)(1-nu/N)^N(1-nu/N)^(-n)
(5)
=lim_(N->infty)(N(N-1)...(N-n+1))/(N^n)(nu^n)/(n!)(1-nu/N)^N(1-nu/N)^(-n)
(6)
=1·(nu^n)/(n!)·e^(-nu)·1
(7)
=(nu^ne^(-nu))/(n!),
(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 N has completely dropped out of the probability function, which has the same functional form for all values of nu.

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

sum_(n=0)^inftyP_nu(n)=e^(-nu)sum_(n=0)^infty(nu^n)/(n!)=e^(-nu)e^nu=1.
(9)

The ratio of probabilities is given by

(P_nu(n=i+1))/(P(n=i))=((nu^(i+1)e^(-nu))/((i+1)!))/((e^(-nu)nu^i)/(i!))=nu/(i+1).
(10)

The Poisson distribution reaches a maximum when

(dP_nu(n))/(dn)=(e^(-nu)n(gamma-H_n+lnnu))/(n!)=0,
(11)

where gamma is the Euler-Mascheroni constant and H_n is a harmonic number, leading to the transcendental equation

gamma-H_n+lnnu=0,
(12)

which cannot be solved exactly for n.

The moment-generating function of the Poisson distribution is given by

M(t)=e^(-nu)e^(nue^t)=e^(nu(e^t-1))
(13)
M^'(t)=nue^te^(nu(e^t-1))
(14)
M^('')(t)=(nue^t)^2e^(nu(e^t-1))+nue^te^(nu(e^t-1))
(15)
R(t)=nu(e^t-1)
(16)
R^'(t)=nue^t
(17)
R^('')(t)=nue^t,
(18)

so

mu=R^'(0)=nu
(19)
sigma^2=R^('')(0)=nu
(20)

(Papoulis 1984, p. 554).

The raw moments can also be computed directly by summation, which yields an unexpected connection with the Bell polynomial phi_n(x) and Stirling numbers of the second kind,

phi_n(x)=sum_(k=0)^infty(e^(-x)x^k)/(k!)k^n=sum_(k=1)^nx^kS(n,k)
(21)

known as Dobiński's formula. Therefore,

mu_2^'=nu(1+nu)
(22)
mu_3^'=nu(1+3nu+nu^2)
(23)
mu_4^'=nu(1+7nu+6nu^2+nu^3).
(24)

The central moments can then be computed as

mu_2=nu
(25)
mu_3=nu
(26)
mu_4=nu(1+3nu),
(27)

so the mean, variance, skewness, and kurtosis are

mu=nu
(28)
sigma^2=nu
(29)
gamma_1=(mu_3)/(sigma^3)=nu/(nu^(3/2))=nu^(-1/2)
(30)
gamma_2=(mu_4)/(sigma^4)-3=(nu(1+3nu))/(nu^2)-3
(31)
=(nu+3nu^2-3nu^2)/(nu^2)=nu^(-1).
(32)

The characteristic function for the Poisson distribution is

phi(t)=e^(nu(e^(it)-1))
(33)

(Papoulis 1984, pp. 154 and 554), and the cumulant-generating function is

K(h)=nu(e^h-1)=nu(h+1/(2!)h^2+1/(3!)h^3+...),
(34)

so

kappa_r=nu.
(35)

The mean deviation of the Poisson distribution is given by

MD=(2e^(-nu)nu^(|_nu_|+1))/(|_nu_|!).
(36)

The Poisson distribution can also be expressed in terms of

lambda=nu/x,
(37)

the rate of changes, so that

P_nu(n)=((lambdax)^ne^(-lambdax))/(n!).
(38)

The moment-generating function of a Poisson distribution in two variables is given by

M(t)=e^((nu_1+nu_2)(e^t-1)).
(39)

If the independent variables x_1, x_2, ..., x_N have Poisson distributions with parameters mu_1, mu_2, ..., mu_N, then

X=sum_(j=1)^Nx_j
(40)

has a Poisson distribution with parameter

mu=sum_(j=1)^Nmu_j.
(41)

This can be seen since the cumulant-generating function is

K_j(h)=mu_j(e^h-1)
(42)
K=sum_(j)K_j(h)=(e^h-1)sum_(j)mu_j=mu(e^h-1).
(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

f_b(N)=(N^_(1-b))/(N!)[N^_(1-b)+Nb]^(N-1)e^(N^_(1-b)-Nb),
(44)

where N is the number of galaxies in a volume V, N^_=n^_V, n^_ is the average density of galaxies, and b=-W/(2K) approx 0.70+/-0.05, with 0<=b<1 is the ratio of gravitational energy to the kinetic energy of peculiar motions, Letting b=0 gives

f_0(N)=(e^(-N^_)N^_^N)/(N!),
(45)

which is indeed a Poisson distribution with nu=N^_. Similarly, letting b=1 gives f_1(N)=0.

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