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