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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.
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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