Triangular(min,most_likely,max)

Crystal Ball parameter restrictions

The Triangular distribution constructs a triangular shape from the three input parameters. An example of the Triangular distribution is given below:

#### Uses

The Triangular distribution is used as a rough modeling tool where the range (*a* to *c*) and the most likely value (*b*) can be estimated. It has no theoretical basis but derives its statistical properties from its geometry.

The Triangular distribution's considerable flexibility in its shape, coupled with the intuitive nature of its parameters made it quickly gain popularity among risk analysts. However, *a* and *c* are the *absolute* minimum and maximum values, which can be very hard to estimate in practice. As described here, a way to circumvent this problem is to ask the expert for "feasible" or "practical" bounds and the chances of exceeding them, which is essentially asking experts for bounds as percentiles.

Most simulation software offer the possibility to specify the Triangular (and Pert) distributions with percentiles as input parameters, as shown below

It should be noted that the Triangular shape will also usually overemphasize the tails of the distribution and under emphasize the shoulders in comparison with other, more natural, distributions. The PERT distribution takes the same parameters as the Triangular, but generally offers a more reasonable interpretation of the parameter values in modeling expert opinion.

The use of the Triangular in modeling expert opinions is further discussed here.

#### Comments

The sum of two identical independent Uniform distributions is a symmetric Triangular distribution.