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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.

By changing the input parameters of the graphic, you can see the variations in the probability density function, cumulative frequency and characteristics of the triangular distribution.  Clicking the random button will generate 100 random numbers for the current parameters.

Min Mode Max
Random Numbers
60.905 41.298 73.539 43.361 56.985 36.003 63.374 64.756 34.236 57.220 38.221 51.198 45.232 62.692 49.205 29.470 45.659 40.789 47.058 21.960 68.400 43.870 54.303 50.988 56.451 36.292 46.597 38.999 77.028 47.496 53.755 29.927 39.306 62.207 42.459 44.972 50.420 35.176 41.023 35.874 30.398 58.290 57.706 64.339 32.704 49.899 44.451 27.765 53.593 37.286 48.190 66.216 38.869 59.677 31.939 42.152 52.180 45.849 34.748 42.834 54.660 36.773 41.705 51.794 67.680 52.587 58.755 43.075 32.009 48.443 55.445 38.404 28.822 46.148 24.569 70.239 44.085 49.729 39.604 31.223 47.734 66.026 27.617 42.015 60.156 34.442 40.430 39.980 26.394 25.232 37.845 33.702 52.877 61.459 33.094 69.602 72.904 37.532 48.716 55.861

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 cautious 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+sqr(u*(max-min)*(mode-min))
else
  r=max-sqr((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: 07-Nov-2007

 

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