Logarithmic distribution
Probability mass function
The function is only defined at integer values. The connecting lines are merely guides for the eye. |
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Cumulative distribution function
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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 Xi, 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
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Further reading
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- Weisstein, Eric W., "Log-Series Distribution", MathWorld.