Probability Distributions > Triangular Distribution

## What is a Triangular Distribution?

A triangular distribution (sometimes called a triangle distribution) is a continuous probability distribution shaped like a triangle. It is defined by:

- a: the
**minimum value**, where a ≤ c, - c: the
**peak value**(the height of the triangle), where a ≤ c ≤ b, - b: the
**maximum value**, where b ≥ c.

This makes it very easy to estimate the distribution’s parameters from sample data:

- Use the sample minimum as an estimator for a,
- Use the sample maximum as an estimator for b, and
- Use any reasonable statistic (e.g. the sample mean, mode or median) as an estimator for c.

If you don’t have sample data, expert knowledge can be used to estimate a probable minimum, maximum and most likely value (i.e. the mode).

The three parameters, a b and c change the shape of the triangle:

Like all probability distributions, the area under the curve is 1. Therefore, the wider the distance between a and c (i.e. the range), the shorter the height.

When the peak is centered at zero and a = b, it is called a **symmetric triangular distribution**. When this happens, a and b are equal but opposite in sign (e.g. -2, 2) and are sometimes referred to as -a and a instead of a and b.

## PDF, Mean and Standard Deviation

The probability density function, which is used to find the probability a random variable falls into a certain range, is given by:

The mean for this distribution is:

μ = 1/3 (a + b + c).

The standard deviation, s, is:

s = (1/√6) a.

This formula makes the assumption that the distribution is centered at zero and that the endpoints are known.

**Reference**:

Samuel Kotz, S and van Dorp.J. (2004) Beyond Beta. Sample chapter on the Triangle Distribution available here from World Scientific.

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