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.

Probability density function
The probability density function (pdf) for the Lomax distribution is given by
 * $$p(x) = {\alpha \over \lambda} \left[{1 + {x \over \lambda}}\right]^{-(\alpha+1)}, \qquad x \geq 0,$$

with shape parameter $$\alpha > 0$$ and scale parameter $$\lambda > 0$$. The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is:
 * $$p(x) = {{\alpha\lambda^\alpha} \over {(x + \lambda)^{\alpha+1}}}$$.

Non-central moments
The $$\nu$$th non-central moment $$E\left[X^\nu\right]$$ exists only if the shape parameter $$\alpha$$ strictly exceeds $$\nu$$, when the moment has the value
 * $$E\left(X^\nu\right) = \frac{\lambda^\nu \Gamma(\alpha - \nu)\Gamma(1 + \nu)}{\Gamma(\alpha)}$$

Relation to the Pareto distribution
The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically:
 * $$\text{If } Y \sim \mbox{Pareto}(x_m = \lambda, \alpha), \text{ then } Y - x_m \sim \mbox{Lomax}(\alpha,\lambda).$$

The Lomax distribution is a Pareto Type II distribution with xm=λ and μ=0:
 * $$\text{If } X \sim \mbox{Lomax}(\alpha, \lambda) \text{ then } X \sim \text{P(II)}\left(x_m = \lambda, \alpha, \mu = 0\right).$$

Relation to the generalized Pareto distribution
The Lomax distribution is a special case of the generalized Pareto distribution. Specifically:
 * $$\mu = 0,~ \xi = {1 \over \alpha},~ \sigma = {\lambda \over \alpha} .$$

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

Relation to the F distribution
The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density $$f(x) = \frac{1}{(1 + x)^2}$$, the same distribution as an F(2,2) distribution. This is the distribution of the ratio of two independent and identically distributed random variables with exponential distributions.

Relation to the q-exponential distribution
The Lomax distribution is a special case of the q-exponential distribution. The q-exponential extends this distribution to support on a bounded interval. The Lomax parameters are given by:
 * $$\alpha = {{2 - q} \over {q - 1}}, ~ \lambda = {1 \over \lambda_q(q - 1)} .$$

Relation to the (log-) logistic distribution
The logarithm of a Lomax(shape = 1.0, scale = λ)-distributed variable follows a logistic distribution with location log(λ) and scale 1.0. This implies that a Lomax(shape = 1.0, scale = λ)-distribution equals a log-logistic distribution with shape β = 1.0 and scale α = log(λ).

Gamma-exponential (scale-) mixture connection
The Lomax distribution arises as a mixture of exponential distributions where the mixing distribution of the rate is a gamma distribution. If λ|k,θ ~ Gamma(shape = k, scale = θ) and X|λ ~ Exponential(rate = λ) then the marginal distribution of X|k,θ is Lomax(shape = k, scale = 1/θ). Since the rate parameter may equivalently be reparameterized to a scale parameter, the Lomax distribution constitutes a scale mixture of exponentials (with the exponential scale parameter following an inverse-gamma distribution).