# 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.

Parameters Probability density function Cumulative distribution function $\alpha >0$ shape (real) $\lambda >0$ scale (real) $x\geq 0$ ${\alpha \over \lambda }\left[{1+{x \over \lambda }}\right]^{-(\alpha +1)}$ $1-\left[{1+{x \over \lambda }}\right]^{-\alpha }$ ${\lambda \over {\alpha -1}}{\text{ for }}\alpha >1$ Otherwise undefined $\lambda ({\sqrt[{\alpha }]{2}}-1)$ 0 ${{\lambda ^{2}\alpha } \over {(\alpha -1)^{2}(\alpha -2)}}{\text{ for }}\alpha >2$ $\infty {\text{ for }}1<\alpha \leq 2$ Otherwise undefined ${\frac {2(1+\alpha )}{\alpha -3}}\,{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3\,$ ${\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4\,$ ## Characterization

### 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 $E[X^{\nu }]$ exists only if the shape parameter $\alpha$ strictly exceeds $\nu$ , when the moment has the value

$E(X^{\nu })={\frac {\lambda ^{\nu }\Gamma (\alpha -\nu )\Gamma (1+\nu )}{\Gamma (\alpha )}}$ ### 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:

${\text{If }}X\sim {\mbox{Lomax}}(\alpha ,\lambda ){\text{ then }}X\sim {\text{P(II)}}(x_{m}=\lambda ,\alpha ,\mu =0).$ ### 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 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).

## See also

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