Lomax distribution

The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.

The probability density function (pdf) for the Lomax distribution is given by

with shape parameter ${\displaystyle \alpha >0}$ and scale parameter ${\displaystyle \lambda >0}$. The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is:

The ${\displaystyle \nu }$th non-central moment ${\displaystyle E\left[X^{\nu }\right]}$ exists only if the shape parameter ${\displaystyle \alpha }$ strictly exceeds ${\displaystyle \nu }$, when the moment has the value

The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically:

The Lomax distribution is a Pareto Type II distribution with x_m = λ and μ = 0:

The Lomax distribution is a special case of the generalized Pareto distribution. Specifically:

The Lomax distribution with scale parameter λ = 1 is a special case of the beta prime distribution. If X has a Lomax distribution, then ${\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}$.