# Logarithmic distribution

Probability mass function The function is only defined at integer values. The connecting lines are merely guides for the eye. | |

Cumulative distribution function | |

Parameters | |
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Support | |

pmf | |

CDF | |

Mean | |

Mode | |

Variance | |

MGF | |

CF | |

PGF |

In probability and statistics, the **logarithmic distribution** (also known as the **logarithmic series distribution** or the **log-series distribution**) is a discrete probability distribution derived from the Maclaurin series expansion

From this we obtain the identity

This leads directly to the probability mass function of a Log(*p*)-distributed random variable:

for *k* ≥ 1, and where 0 < *p* < 1. Because of the identity above, the distribution is properly normalized.

The cumulative distribution function is

where *B* is the incomplete beta function.

A Poisson compounded with Log(*p*)-distributed random variables has a negative binomial distribution. In other words, if *N* is a random variable with a Poisson distribution, and *X*_{i}, *i* = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log(*p*) distribution, then

has a negative binomial distribution. In this way, the negative binomial distribution is seen to be a compound Poisson distribution.

R. A. Fisher described the logarithmic distribution in a paper that used it to model relative species abundance.^{[1]}

The probability mass function *ƒ* of this distribution satisfies the recurrence relation

## See also

- Poisson distribution (also derived from a Maclaurin series)

## References

- ↑ Fisher, R. A.; Corbet, A. S.; Williams, C. B. (1943). "The Relation Between the Number of Species and the Number of Individuals in a Random Sample of an Animal Population" (PDF).
*Journal of Animal Ecology*.**12**(1): 42–58. doi:10.2307/1411. JSTOR 1411.

## Further reading

- Johnson, Norman Lloyd; Kemp, Adrienne W; Kotz, Samuel (2005). "Chapter 7: Logarithmic and Lagrangian distributions".
*Univariate discrete distributions*(3 ed.). John Wiley & Sons. ISBN 978-0-471-27246-5. - Weisstein, Eric W. "Log-Series Distribution".
*MathWorld*.