Logarithmic distribution

{{Probability distribution|

name =Logarithmic|

type =mass|

pdf_image =Image:Logarithmicpmf.svgThe function is only defined at integer values. The connecting lines are merely guides for the eye. |

cdf_image =Image:Logarithmiccdf.svg|

parameters = $0 < p < 1$ |

support = $k \in \{1,2,3,\ldots\}$ |

pdf = $\frac{-1}{\ln(1-p)} \frac{p^k}{k}$ |

cdf = $1 + \frac{\Beta(p;k+1,0)}{\ln(1-p)}$ |

mean = $\frac{-1}{\ln(1-p)} \frac{p}{1-p}$ |

median =|

mode = $1$ |

variance = $- \frac{p^2 + p\ln(1-p)}{(1-p)^2(\ln(1-p))^2}$ |

skewness =|

kurtosis =|

entropy =|

mgf = $\frac{\ln(1 - pe^t)}{\ln(1-p)}\text{ for }t < -\ln p$ |

char = $\frac{\ln(1 - pe^{it})}{\ln(1-p)}$ |

pgf = $\frac{\ln(1-pz)}{\ln(1-p)}\text{ for }|z| < \frac{1}{p}$ |

}}

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

: $-\ln(1-p) = p + \frac{p^2}{2} + \frac{p^3}{3} + \cdots.$

From this we obtain the identity

: $\sum_{k=1}^{\infty} \frac{-1}{\ln(1-p)} \; \frac{p^k}{k} = 1.$

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

: $f(k) = \frac{-1}{\ln(1-p)} \; \frac{p^k}{k}$

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

The cumulative distribution function is

: $F(k) = 1 + \frac{\Beta(p; k+1,0)}{\ln(1-p)}$

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

: $\sum_{i=1}^N X_i$

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.{{Cite journal

|doi = 10.2307/1411

|title = The Relation Between the Number of Species and the Number of Individuals in a Random Sample of an Animal Population

|jstor = 1411

|url = http://www.math.mcgill.ca/~dstephens/556/Papers/Fisher1943.pdf

|year = 1943

|journal = Journal of Animal Ecology

|pages = 42–58

|volume = 12

|issue = 1

|last1 = Fisher

|first1 = R. A.

|last2 = Corbet

|first2 = A. S.

|last3 = Williams

|first3 = C. B.

|bibcode = 1943JAnEc..12...42F

|url-status = dead

|archive-url = https://web.archive.org/web/20110726144520/http://www.math.mcgill.ca/~dstephens/556/Papers/Fisher1943.pdf

|archive-date = 2011-07-26

}}

Logarithmic distribution

See also

References

Further reading