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.