Triangular Distribution

The Triangular Distribution is typically used as a subjective description of a population for which there is only limited sample data.  It is based on a knowledge of the minimum and maximum and an inspired guess as to what the modal value might be.  Despite being a simplistic description of a population, it is a very useful distribution for modeling processes where the relationship between variables is known, but data is scarce (possibly because of the high cost of collection).

It is also used as an alternative to the Beta distribution in PERT, CPM and similar forms of project management tool.  The section on the Beta distribution contains an example using both the Beta and Triangular distributions.

Profile

Parameters

Parameter	Description	Characteristics
min	Minimum value	A float > -∞ and <= mode
mode	Modal Value	A float >= min or <= max
max	Maximum value	A float >= mode and < ∞

Range

The range is determined by the min and max parameters.

Functions

Properties

Example

Triangular distributions are used in oil and gas exploration where data is expensive to collect and it is almost impossible to model the population being sampled accurately, thus subjectivity plays a greater role than in data rich sectors.  The example below shows a subjective assessment of the expected size of discoveries in the United Kingdom's East Midland's Basin:

See the section on the lognormal distribution for an alternative model.

Parameter Estimation

The parameters of a triangular distribution can be derived directly from the dataset which it is intended to describe or model.  Provided the dataset does not contain any anomalous points, the minimum and maximum can be obtained by sorting the values in ascending order and selecting the first and last points.  Un less the mode is being set subjectively, there are a number of ways of determining the mode including:

Use an algorithm such successive bisection

Use the Max, Min and Mean to estimate the mode

This example illustrates both approaches.  The figures below are the porosity of rock samples from deep bore holes. These are very expensive to collect and only a few are available to the analyst who may be asked to provide input for some form of Monte-Carlo evaluation:

10.0%, 13.5%, 15.5%, 20.0%

The successive bisection algorithm estimates the mode as 14.5%.  The mean value of the observations is 14.75%, using this value together with the min and max into this formula sets the mode as 14.25%:

A subjective estimate of the mode by a cautions analyst might set the mode at 12%.

Random Number Generation

Random number generation (referred to as R) for a triangular distribution can be performed by transforming a continuous uniform variable in the range 0 to 1 (referred to as U) with the distribution's inverse probability function:

r=g(u)

Using Basic style code, the function would be similar to:

u=rnd()

if u <= (mode-min)/(max-min) then
  r=min+sqrt(u*(max-min)*(mode-min))
else
  r=max-sqrt((1-u)*(max-min)*(max-mode))
end if

Storing the value from the random number function in the variable u is important because most random number function return a new value each time they are called.  Without the use of the u variable, the statement would use one value for branching and another for calculation.

Page updated: 20-Jun-2008

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