probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a

discrete probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

derived from the Maclaurin series expansion :

-\ln(1-p)  = p + \frac + \frac + \cdots.

From this we obtain the identity :

\sum_^ \frac \; \frac = 1.

This leads directly to the

probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...

of a Log(''p'')-distributed

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

: :

f(k) = \frac \; \frac

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

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...

is :

F(k) = 1 + \frac

where ''B'' is the

incomplete beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t ...

. A Poisson compounded with Log(''p'')-distributed random variables has a

negative binomial distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expr ...

. In other words, if ''N'' is a random variable with a

Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known ...

, 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_^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.

See also

References

Further reading