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
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
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
where is the floor function
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