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Geometric Distribution


GeometricDistribution

The geometric distribution is a discrete distribution for n=0, 1, 2, ... having probability density function

P(n)=p(1-p)^n
(1)
=pq^n,
(2)

where 0<p<1, q=1-p, and distribution function is

D(n)=sum_(k=0)^(n)P(k)
(3)
=1-q^(n+1).
(4)

The geometric distribution is the only discrete memoryless random distribution. It is a discrete analog of the exponential distribution.

Note that some authors (e.g., Beyer 1987, p. 531; Zwillinger 2003, pp. 630-631) prefer to define the distribution instead for n=1, 2, ..., while the form of the distribution given above is implemented in the Wolfram Language as GeometricDistribution[p].

P(n) is normalized, since

 sum_(n=0)^inftyP(n)=sum_(n=0)^inftyq^np=psum_(n=0)^inftyq^n=p/(1-q)=p/p=1.
(5)

The raw moments are given analytically in terms of the polylogarithm function,

mu_k^'=sum_(n=0)^(infty)P(n)n^k
(6)
=sum_(n=0)^(infty)p(1-p)^nn^k
(7)
=pLi_(-k)(1-p).
(8)

This gives the first few explicitly as

mu_1^'=(1-p)/p
(9)
mu_2^'=((2-p)(1-p))/(p^2)
(10)
mu_3^'=((1-p)[6+(p-6)p])/(p^3)
(11)
mu_4^'=((2-p)(1-p)[12+(p-12)p])/(p^4).
(12)

The central moments are given analytically in terms of the Lerch transcendent as

mu_k=sum_(n=0)^(infty)P(n)(n-(1-p)/p)^k
(13)
=pPhi(1-p,-k,(p-1)/p).
(14)

This gives the first few explicitly as

mu_2=(1-p)/(p^2)
(15)
=q/(p^2)
(16)
mu_3=((p-1)(p-2))/(p^3)
(17)
=(q(2-p))/(p^3)
(18)
mu_4=((p-1)(-p^2+9p-9))/(p^4),
(19)

so the mean, variance, skewness, and kurtosis excess are given by

mu=(1-p)/p
(20)
sigma^2=(1-p)/(p^2)
(21)
gamma_1=(2-p)/(sqrt(1-p))
(22)
gamma_2=(p^2-6p+6)/(1-p).
(23)

For the case p=1/2 (corresponding to the distribution of the number of coin tosses needed to win in the Saint Petersburg paradox) the formula (23) gives

 mu_k^'|_(p=1/2)=1/2Li_(-k)(1/2).
(24)

The first few raw moments are therefore 1, 3, 13, 75, 541, .... Two times these numbers are OEIS A000629, which have exponential generating functions f(x)=-ln(2-e^x) and g(x)=e^x/(2-e^x). The mean, variance, skewness, and kurtosis excess of the case p=q=1/2 are given by

mu=1
(25)
sigma^2=2
(26)
gamma_1=3/2sqrt(2)
(27)
gamma_2=(13)/2.
(28)

The characteristic function is given by

 phi(t)=p/(1-(1-p)e^(it)).
(29)

The first cumulant of the geometric distribution is

 kappa_1=(1-p)/p,
(30)

and subsequent cumulants are given by the recurrence relation

 kappa_(r+1)=(p-1)(dkappa_r)/(dp).
(31)

The mean deviation of the geometric distribution is

 MD=2(1-p)^(|_1/p_|)|_1/p_|,
(32)

where |_x_| is the floor function.


See also

Geometric Series, Hypergeometric Distribution, Saint Petersburg Paradox

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 531-532, 1987.Sloane, N. J. A. Sequence A000629 in "The On-Line Encyclopedia of Integer Sequences."Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 118, 1992.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st ed. Boca Raton, FL: CRC Press, pp. 630-631, 2003.

Referenced on Wolfram|Alpha

Geometric Distribution

Cite this as:

Weisstein, Eric W. "Geometric Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GeometricDistribution.html

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