Laha Distribution

< List of probability distributions

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 Cauchy’s 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.

This special type of random variable follows an ‘absolutely continuous’ Lebesgue measure with respect to specific parameters and it is defined by a particular form of probability density – making it subtly distinct from other non-normal distributions.

Laha distribution properties

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] also notes a contradiction in the literature [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.