A distribution such that

(1) |

(2) |

(3) |

(4) | |||

(5) | |||

(6) | |||

(7) |

Therefore,

(8) | |||

(9) | |||

(10) | |||

(11) |

and

(12) | |||

(13) | |||

(14) |

so the Mean, Variance, Skewness, and Kurtosis are given by

(15) | |||

(16) | |||

(17) | |||

(18) |

In fact, the moments of the distribution are given analytically in terms of the Polylogarithm function,

(19) |

(20) | |||

(21) | |||

(22) | |||

(23) |

so the Mean, Variance, Skewness, and Kurtosis in this case are

(24) | |||

(25) | |||

(26) | |||

(27) |

The first Cumulant of the geometric distribution is

(28) |

(29) |

**References**

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, pp. 531-532,
1987.

Spiegel, M. R. *Theory and Problems of Probability and Statistics.*
New York: McGraw-Hill, p. 118, 1992.

© 1996-9

1999-05-25