Brighton Webs Ltd
statistics for energy and the environment
The Triangular Distribution is often used as a
subjective description of a population. It is based on estimates 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 useful distribution for
modeling subjective probability where hard data
is scarce (possibly because of the high cost of collection). One way of obtaining the the min, mode and max parameters is to ask the opinion
of someone with appropriate experience. The human mind is good at reducing
complex datasets to managable proportions.
It can be 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
Relative to distributions such as the normal and lognormal where only a small
proportion of the population is located in the tails, in some applications, the
triangular distribution can return a higher proportion of values which are close
to the mininum and maximum values.
||A float > -∞ and <= mode
||A float >= min or <= max
||A float >= mode and < ∞
The range is determined by the min and max parameters.
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 graph below illustrates the response you might get from a
geologist if you asked the question what are minimum, most likely and maximum
discovery sizes for a given play.
See the section on the lognormal distribution for an alternative model.
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
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
A subjective estimate of the mode by a cautious
analyst might set the mode at 12%.
Random number generation 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:
The code in the frame below performs a simple simulation of a triangularly
distributed variable. It generates two lists, one contained the simulated
values and the other the expected count for the same interval.
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
Python is sensitive to indentation errors, it maybe necessary to perform some
adjustment of the code following a copy and paste operation.