Definition:Logarithmic Distribution

Definition

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

$X$ has the logarithmic distribution if it has probability density function:

$\map \Pr {X = k} = -\dfrac {\theta^k} {k \ln \paren {1 - \theta} }$

where:

$k = 1, 2, \ldots$

$\theta \in \openint 0 1$

Also known as

The logarithmic distribution is also known as the log-series distribution.

Also see

• Results about the logarithmic distribution can be found here.

Historical Note

The logarithmic distribution was designed by Ronald Aylmer Fisher in $1941$.

It is widely used in studies of species diversity.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): logarithmic distribution