The distribution simplifies when c = a or c = b. For example, if a = 0, b = 1 and c = 1, then the
PDF and
CDF become:
Distribution of the absolute difference of two standard uniform variables
This distribution for a = 0, b = 1 and c = 0 is the distribution of X = |X1 − X2|, where X1, X2 are two independent random variables with standard
uniform distribution.
Symmetric triangular distribution
The symmetric case arises when c = (a + b) / 2.
In this case, an alternate form of the distribution function is:
Distribution of the mean of two standard uniform variables
This distribution for a = 0, b = 1 and c = 0.5—the mode (i.e., the peak) is exactly in the middle of the interval—corresponds to the distribution of the mean of two standard uniform variables, that is, the distribution of X = (X1 + X2) / 2, where X1, X2 are two independent random variables with standard
uniform distribution in [0, 1].[1] It is the case of the
Bates distribution for two variables.
Generating random variates
Given a random variate U drawn from the
uniform distribution in the interval (0, 1), then the variate
The triangular distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known but data is scarce (possibly because of the high cost of collection).
It is based on a knowledge of the minimum and maximum and an "inspired guess"[3] as to the modal value. For these reasons, the triangle distribution has been called a "lack of knowledge" distribution.
Business simulations
The triangular distribution is therefore often used in
business decision making, particularly in
simulations. Generally, when not much is known about the
distribution of an outcome (say, only its smallest and largest values), it is possible to use the
uniform distribution. But if the most likely outcome is also known, then the outcome can be simulated by a triangular distribution. See for example under
corporate finance.
Project management
The triangular distribution, along with the
PERT distribution, is also widely used in
project management (as an input into
PERT and hence
critical path method (CPM)) to model events which take place within an interval defined by a minimum and maximum value.
Audio dithering
The symmetric triangular distribution is commonly used in
audio dithering, where it is called TPDF (triangular probability density function).
Central limit theorem — The triangle distribution often occurs as a result of adding two uniform random variables together. In other words, the triangle distribution is often (not always) the result of the first iteration of the central limit theorem summing process (i.e. ). In this sense, the triangle distribution can occasionally occur naturally. If this process of summing together more random variables continues (i.e. ), then the distribution will become increasingly bell-shaped.
Irwin–Hall distribution — Using an Irwin–Hall distribution is an easy way to generate a triangle distribution.
Bates distribution — Similar to the Irwin–Hall distribution, but with the values rescaled back into the 0 to 1 range. Useful for computation of a triangle distribution which can subsequently be rescaled and shifted to create other triangle distributions outside of the 0 to 1 range.
^"Archived copy"(PDF). www.asianscientist.com. Archived from
the original(PDF) on 7 April 2014. Retrieved 12 January 2022.{{
cite web}}: CS1 maint: archived copy as title (
link)