The geometric distribution is a discrete distribution for , 1, 2, ... having probability density function
(1)
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(2)
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where , , and distribution function is
(3)
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(4)
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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
(5)
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The raw moments are given analytically in terms of the polylogarithm function,
(6)
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(7)
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(8)
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This gives the first few explicitly as
(9)
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(10)
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(11)
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(12)
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The central moments are given analytically in terms of the Lerch transcendent as
(13)
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(14)
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This gives the first few explicitly as
(15)
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(16)
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(17)
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(18)
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(19)
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so the mean, variance, skewness, and kurtosis excess are given by
(20)
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(21)
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(22)
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(23)
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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
(24)
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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
(25)
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(26)
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(27)
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(28)
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The characteristic function is given by
(29)
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The first cumulant of the geometric distribution is
(30)
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and subsequent cumulants are given by the recurrence relation
(31)
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The mean deviation of the geometric distribution is
(32)
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where is the floor function.