Laha Distribution

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The Laha distribution is a continuous, unimodal (one peak) and univariate probability distribution with infinite support.

In 1958, the Laha distribution was introduced as a unique example of its kind; one that abides by the Cauchy law in terms of probability quotients. Laha originally formulated the distribution to disprove the belief that two IID random variables is distributed as Cauchy if the distribution is normal.

Laha distribution properties

This special type of random variable follows an absolutely continuous Lebesgue measure with respect to specific parameters, which means that it is possible to integrate the function over any interval. Functions that are absolutely continuous in the real-numbered domain are those that adhere to the Fundamental Theorem of Calculus. A real-valued random variable X is absolutely continuous if its distribution function is absolutely continuous.

A random variable X has a Laha distribution L (0; 1) if its distribution is absolutely continuous with respect to the Lebesgue measure, with probability density [1]

A random variable X has a Laha distribution L(μ, σ2) with parameters μ ∈ ℝ,σ2, if its probability density is

laha distribution

The Laha is a special case of the generalized Pearson VII distribution and is a 4th order Grand Unified distribution [2].

Crooks [2] notes a contradiction in the literature — there are actually two ways to generate Laha random variables [1]:

Laha random variates can be easily generated by noting that the distribution is symmetric, and that the half-Laha distribution is a special case of the generalized beta prime distribution, which can itself be generated as the ratio of two gamma distributions [1].

Crooks, G. E. [2]


[1] Popescu, I. & Dumitrescu, M. Laha Distribution: Computer Generation and Applications to Life Time Modelling. Journal of Universal Computer Science, vol. 5, no. 8 (1999), 471-48

[2] Crooks, G. Field Guide to Continuous Probability Distributions.

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