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. 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. The probability density function (pdf) for the Lomax distribution is given by with shape parameter α > 0 {displaystyle alpha >0} and scale parameter λ > 0 {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: The ν {displaystyle u } th non-central moment E [ X ν ] {displaystyle E} exists only if the shape parameter α {displaystyle alpha } strictly exceeds ν {displaystyle u } , when the moment has the value 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 xm=λ and μ=0: The Lomax distribution is a special case of the generalized Pareto distribution. Specifically: The Lomax distribution with scale parameter λ = 1 is a special case of the beta prime distribution. If X has a Lomax distribution, then X λ ∼ β ′ ( 1 , α ) {displaystyle {frac {X}{lambda }}sim eta ^{prime }(1,alpha )} . The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density f ( x ) = 1 ( 1 + x ) 2 {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.