Logarithmic distribution

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

<math>

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

</math>

From this we obtain the identity

<math>\sum_{k=1}^{\infty} \frac{-1}{\ln(1-p)} \; \frac{p^k}{k} = 1. </math>

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

<math> f(k) = \frac{-1}{\ln(1-p)} \; \frac{p^k}{k}</math>

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

The cumulative distribution function is

<math> F(k) = 1 + \frac{\Beta(p; k+1,0)}{\ln(1-p)}</math>

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

<math>\sum_{i=1}^N X_i</math>

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.<ref>Template:Cite journal</ref>

See also

• Poisson distribution (also derived from a Maclaurin series)

References

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Further reading

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