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For experienced modelers, the most challenging task in creating a
simulation model is usually not identifying the key inputs and
outputs, but selecting an appropriate probability distribution
and parameters to model the uncertainty of each input
variable. For example, Risk Solver software provides over 40
analytic probability distributions -- which one should you use?
The answer depends on your application, but some general guidelines
can be given.
If you must choose or create your own
distribution, the first step is to determine whether to use a
discrete or continuous form.
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If there are a small
number of possible values for the uncertain variable, you may be
able to use a discrete analytic distribution, or construct a
discrete custom distribution. If the underlying physical
process involves discrete, countable entities -- such as the number
of customers arriving at a service window -- you can use a discrete
distribution. |
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If the possible values are highly divisible --
such as most prices, volumes, interest rates, exchange rates,
weights, distances, etc. -- you will likely use a continuous
distribution. In some cases, you may use a continuous
distribution to approximate a discrete distribution.
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Another characteristic that distinguishes
probability distributions is the range of sample values they can generate.
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Some distributions
are intrinsically bounded -- samples are guaranteed to
lie between a known minimum and maximum value. Examples
are the Uniform, Triangular, Beta, and Binomial distributions. |
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Other analytic
distributions are unbounded -- sample values may cluster
around the distribution’s mean, but may sometimes have extreme
negative or positive values. Examples are the Normal,
Logistic, and Extreme Value distributions. |
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Still other
distributions are partially bounded, with a known minimum
such as zero, but no maximum value. Examples are the
Exponential, Poisson, and Weibull distributions. |
At times, you may find that the most appropriate
distribution (say the Normal) is unbounded, but you know that the
realistic values of the physical process are bounded, or your model is
designed to handle values only up to some realistic limit. Your
software may allow you to truncate an unbounded distribution. For
example, in Risk Solver you can
impose bounds on any distribution by passing the PsiTruncate
property function as an argument to the distribution function.
A third characteristic of probability
distributions is whether they are analytic (also called
parametric) or custom (sometimes called non-parametric)
distributions.
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An analytic
distribution has a form derived from certain theoretical
assumptions about the problem. For example, a Poisson
distribution is derived from an assumption that events are
independent and occur at a known average rate, and an Exponential
distribution is derived from an assumption of a constant rate of
decay in some process. |
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A custom
distribution has a form dictated by either past data or expert
opinion about the range and frequency of sample values.
Risk Solver software provides
five general-purpose functions -- PsiCumul, PsiDiscrete,
PsiDisUniform, PsiGeneral and PsiHistogram -- to help you model
custom distributions. |
Generally speaking, you should choose an analytic
distribution if -- and only if -- the theoretical assumptions truly
apply in your situation.
Using a Triangular Distribution. If
you have only estimates of the minimum, maximum, and most likely
values of an uncertain variable -- and no other past data or
literature references -- a popular approach is to create a
Triangular distribution from these three numbers. This is
unlikely to be a highly accurate representation of the
uncertainty, but it will allow you to get started, and it is far
better than a single average that is subject to the Flaw of
Averages. If your ‘minimum’ and ‘maximum’ values are really
low- and high-percentile estimates rather than the absolute
lowest and highest values that can occur, consider using a
'generalized Triangular' distribution (PsiTriangGen in Risk Solver)
instead. Define Each Uncertain Variable
Only Once. Often, you’ll need to use the same uncertain
variable in several different formulas in your model. A very
common error is to enter the same distribution function, with the
same parameters (say PsiNormal(100, 10) in Risk Solver),
several times in a model -- in a belief that these instances will
yield the same results on each trial. This is incorrect
-- by doing this, you’ve actually defined several independent
uncertain variables that may well sample different values on each
trial. You should instead define =PsiNormal(100, 10) only once
(for example in a cell such as A1), and use A1 in every formula
where the uncertain variable is needed.
Choosing a Distribution
Back to Monte
Carlo Simulation Tutorial
Back to Simulation Introduction |
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