Create a distribution from a set of points on a curve

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Situation

We have a set of co-ordinates that we wish to use to construct a distribution:

1. {x, f(x)} for a continuous distribution where f(x) is (or is proportional to) the probability density at value x;

2. {x, F(x)} a continuous distribution where F(x) is the cumulative probability (P(X<=x)) at value x; or

3. {x, p(x)} for a discrete distribution where p(x) is (or is proportional to) the probability of value x.

Uses

There are many uses of this technique. For example:

• Converting the results of a constructed Bayesian inference calculation into a distribution; or
• knowing the pdf or pmf of a specific distribution not available with ModelRisk.

Application

We can use the same techniques as explained in Method 3 to create distributions from a set of points:

If the data set is of the form of {x, f(x)}, we can use the VoseRelative function in ModelRisk; 

If the data set is of the form {x, F(x)}, we can use the VoseCumulA (or VoseCumulD) function in ModelRisk; or

If the data set is of the form {x, p(x)}, we can use the VoseDiscrete function in ModelRisk.

The three functions have similar formats:

=VoseRelative(min,max,{x},{f(x)})

=VoseCumulA(min,max,{x},{F(x)})

=VoseDiscrete({x},{p(x)})

The {x} values must be in ascending order for the VoseRelative and VoseCumulA functions because they construct a distribution shape. For the VoseDiscrete function this is unnecessary because it is simply a list of values.