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. See also Poisson distribution (also derived from a Maclaurin series) References Further reading Discrete distributions Logarithms