GeometricDistribution

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

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

The geometric appropriation is the main discrete memoryless irregular conveyance. It is a discrete sample of the exponential dispersion. 

Note that a few creators (e.g., Beyer 1987, p. 531; Zwillinger 2003, pp. 630-631) want to characterize the dissemination rather for n=1, 2, …, while the type of the circulation given above is executed 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.

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




So the first few explicitly as





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



the mean, variance, skewness, and kurtosis excess are





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).

The initial barely any crude minutes are along these lines 1, 3, 13, 75, 541, …. Multiple times these numbers are OEIS A000629, which have exponential creating capacities f(x)=-ln(2-e^x) and g(x)=e^x/(2-e^x). The mean, difference, skewness, and kurtosis abundance of the case p=q=1/2 are given by





The characteristic function is given by

The first cumulant  of the geometric distribution is

and subsequent cumulants are given by the recurrence relation

The mean deviation  of the geometric distribution is


|_x_|

where  is the floor function