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See also: Monte Carlo simulation introduction
Consider the distribution of an uncertain input variable x. The cumulative distribution function F(x) gives the probability P that the variable X will be less than or equal to x, i.e.
F(x) = P(X<=x)
F(x) obviously ranges from zero to one. Now, we can look at this equation in the reverse direction: what is the value of x for a given value of F(x)? This inverse function G(F(x)) is written as:
G(F(x)) = x
It is this concept of the inverse function G(F(x)) that is used in the generation of random samples from each distribution in a risk analysis model. The figure below provides a graphical representation of the relationship between F(x) and G(F(x)):
To generate a random sample for a probability distribution, a Uniform(0,1) random number (U) is generated. This value is then fed into the equation to determine the value to be generated for the distribution:
G(U) = x
The random number U is generated from a Uniform(0,1) distribution to provide equal opportunity of an x value being generated in any percentile range. The inverse function concept is employed in a number of sampling methods, discussed in the following sections. ModelRisk employs this method for all its univariate distributions via the optional U parameter, which then allows you to make maximal use of your Monte Carlo Excel add-in. There is no analogy to the inverse generation method for multivariate distributions.
Read on: Monte Carlo sampling