# The other Burr distributions

As I mentioned in the previous post, there are 12 distributions named for Irving Burr, known as Burr Type I, Burr Type II, Burr Type III, …, Burr Type XII. [1]

The last of these is by far the most common, and the rest are hard to find online. I did manage to find them, so I’ll list them here for my future reference and for the benefit of anyone else interested in these distributions in the future.

Each distribution has a closed-form CDF because each is defined by its CDF. In all but one case, Burr Type XI, the CDF functions are invertible in closed-form. This means that except for Burr Type XI, one can easily generate random samples from each of the Burr distributions by applying the inverse CDF to a uniform random variable.

For each distribution I’ll give the CDF.

Burr Type I distribution

for 0 < x < 1, which is the uniform distribution.

Burr Type II distribution

for -∞ < x < ∞.

Burr Type III distribution

for 0 < x < ∞.

The Burr Type III distribution is also known as the Dagum distribution, and is probably the most well known of the Burr distributions after Type XII.

Burr Type IV distribution

for 0 < x < c.

Burr Type V distribution

for -π/2 < x < π/2.

Burr Type VI distribution

for -∞ < x < ∞.

Burr Type VII distribution

for -∞ < x < ∞.

Burr Type VIII distribution

for -∞ < x < ∞.

Burr Type IX distribution

for -∞ < x < ∞.

Burr Type X distribution

for 0 ≤ x < ∞.

Burr Type XI distribution

for 0 < x < 1.

Burr Type XII distribution

for 0 ≤ x < ∞.

The Burr Type XII distribution is also known as the Singh-Maddala distribution in economics.

[1] It’s possible to represent the Roman numerals I through XII as single Unicode characters as described here. So if you want to get fancy, we have Burr Ⅰ, Burr Ⅱ, Burr Ⅲ, …, Burr Ⅻ. Here Ⅻ, for example, is a single character, U+216B.