Logarithmic Distribution

The logarithmic distribution is a discrete distribution over the positive integers defined by a single parameter \(p \in (0, 1)\).

Definition

• Probability Mass Function for \(x \in \{1, 2, 3, ...\}\)

\[P(X=x; p) = \frac{-1}{\log (1 - p)} \frac{p^x}{x}\]

Derivation

The logarithmic distribution obtains its name from its construction from the Taylor series expansion of the natural logarithm function. Specifically, we start with the Maclaurin series

\[-\log(1 - p) = \sum_{k=1}^{\infty} \frac{p^k}{k}\]

Dividing both sides by \(-\log(1-p)\) gives us a properly normalized probability distribution.

Relation to Other Distributions

Negative Binomial

The Negative Binomial