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

${\displaystyle p(x)={\alpha \over \lambda }\left[{1+{x \over \lambda }}\right]^{-(\alpha +1)},\qquad x\geq 0,}$ 

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:

${\displaystyle p(x)={{\alpha \lambda ^{\alpha }} \over {(x+\lambda )^{\alpha +1}}}}$ .

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

${\displaystyle E\left(X^{\nu }\right)={\frac {\lambda ^{\nu }\Gamma (\alpha -\nu )\Gamma (1+\nu )}{\Gamma (\alpha )}}}$ 

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

${\displaystyle {\text{If }}Y\sim {\mbox{Pareto}}(x_{m}=\lambda ,\alpha ),{\text{ then }}Y-x_{m}\sim {\mbox{Lomax}}(\alpha ,\lambda ).}$ 

The Lomax distribution is a Pareto Type II distribution with x_m=λ and μ=0:^[5]

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

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

${\displaystyle \mu =0,~\xi ={1 \over \alpha },~\sigma ={\lambda \over \alpha }.}$ 

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

The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density ${\displaystyle f(x)={\frac {1}{(1+x)^{2}}}}$ , 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.

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:

${\displaystyle \alpha ={{2-q} \over {q-1}},~\lambda ={1 \over \lambda _{q}(q-1)}.}$ 

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).

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).

• power law
• compound probability distribution
• hyperexponential distribution (finite mixture of exponentials)
• normal-exponential-gamma distribution (a normal scale mixture with Lomax mixing distribution)