Triangle distribution | Vose Software

Triangle distribution

Format: Triangle(min, mode, max)


The Triangle distribution (also known as the Triangular distribution or the Triang distribution) constructs a Triangle shape from its three input parameters. An example of the Triangle distribution is given below:


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

The Triangle distribution offers considerable flexibility in its shape, coupled with the intuitive nature of its defining parameters and speed of use. It has therefore achieved a great deal of popularity among risk analysts. However, a and c are the absolute minimum and maximum estimated values for the variable and it is generally a difficult task to make estimates of these values.

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


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

ModelRisk functions added to Microsoft Excel for the Triangle distribution

VoseTriangle generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseTriangleObject constructs a distribution object for this distribution.

VoseTriangleProb returns the probability density or cumulative distribution function for this distribution.

VoseTriangleProb10 returns the log10 of the probability density or cumulative distribution function.  


Triangle distribution equations