To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time. 4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds # Lomax distribution

Parameters Probability density function Cumulative distribution function $\alpha >0$ shape (real)$\lambda >0$ scale (real) $x\geq 0$ ${\alpha \over \lambda }\left[{1+{x \over \lambda }}\right]^{-(\alpha +1)}$ $1-\left[{1+{x \over \lambda }}\right]^{-\alpha }$ ${\lambda \over {\alpha -1}}{\text{ for }}\alpha >1$ ; undefined otherwise $\lambda \left({\sqrt[{\alpha }]{2}}-1\right)$ 0 ${\begin{cases}{{\lambda ^{2}\alpha } \over {(\alpha -1)^{2}(\alpha -2)}}&\alpha >2\\\infty &1<\alpha \leq 2\\{\text{Undefined}}&{\text{otherwise}}\end{cases}}$ ${\frac {2(1+\alpha )}{\alpha -3}}\,{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3\,$ ${\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4\,$ 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.

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

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

### 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).