Logarithmic distribution

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Logarithmic
Probability mass function
Cumulative distribution function
Parameters
Support
Probability mass function (pmf)
Cumulative distribution function (cdf)
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

In probability and statistics, the logarithmic distribution (also known as the logarithmic 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 , and where . Because of the identity above, the distribution is properly normalized.

The cumulative distribution function is

where is the incomplete beta function.

A Poisson mixture of Log(p)-distributed random variables has a negative binomial distribution. In other words, if is a random variable with a Poisson distribution, and , = 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 applied this distribution to population genetics.

See also

de:Logarithmische Verteilung it:Variabile casuale logaritmica


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