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.^{[1]}^{[2]}^{[3]} It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.^{[4]}

## Characterization

### Probability density function

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

with shape parameter and scale parameter . The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is:

- .

### Non-central moments

The th non-central moment exists only if the shape parameter strictly exceeds , when the moment has the value

## 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:

The Lomax distribution is a Pareto Type II distribution with *x*_{m}=λ and μ=0:^{[5]}

### Relation to the generalized Pareto distribution

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

### 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 .

### Relation to the F distribution

The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density , 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:

### 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).

## See also

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

## References

**^**Lomax, K. S. (1954) "Business Failures; Another example of the analysis of failure data".*Journal of the American Statistical Association*, 49, 847–852. JSTOR 2281544**^**Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). "20*Pareto distributions*".*Continuous univariate distributions*.**1**(2nd ed.). New York: Wiley. p. 573.**^**J. Chen, J., Addie, R. G., Zukerman. M., Neame, T. D. (2015) "Performance Evaluation of a Queue Fed by a Poisson Lomax Burst Process",*IEEE Communications Letters*, 19, 3, 367-370.**^**Van Hauwermeiren M and Vose D (2009).*A Compendium of Distributions*[ebook]. Vose Software, Ghent, Belgium. Available at www.vosesoftware.com.**^**Kleiber, Christian; Kotz, Samuel (2003),*Statistical Size Distributions in Economics and Actuarial Sciences*, Wiley Series in Probability and Statistics,**470**, John Wiley & Sons, p. 60, ISBN 9780471457169.