Geometric Distribution

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

where , , and distribution function is

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 , 2, ..., while the form of the distribution given above is implemented in the Wolfram Language as GeometricDistribution[p].

is normalized, since

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

This gives the first few explicitly as

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

This gives the first few explicitly as

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

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

The first few raw moments are therefore 1, 3, 13, 75, 541, .... Two times these numbers are OEIS A000629, which have exponential generating functions and . The mean, variance, skewness, and kurtosis excess of the case 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.