Probability calculations in ModelRisk | Vose Software

Probability calculations in ModelRisk

ModelRisk has functions for calculating the joint probability density (or probability
mass) f({x}) and joint cumulative probability F({x}) for a set of values {x} against a specified distribution.

These functions offer a simple way of calculating the likelihood of observations being drawn from a specified distribution, which is useful for various statistical models from distribution fitting to hypothesis testing, as well as predicting the likelihood of observing values in the future.

The functions are particularly efficient where you have a large set of values {x} as the required joint probability can be calculated in one single formula. However, the joint probability of probability density for a large set of values can quickly approach values too small for Excel to handle. Therefor ModelRisk has a parallel set of functions that return Log base 10 of the probability calculations.

Probability functions as described below exist for custom distributions constructed through ModelRisk as well: e.g. VoseAggregatePanjerProb, VoseCombinedprob etc.

There are three ModelRisk windows for easily performing probability calculations. These are explained here.

VoseDistributionProb

Calculates the joint probability density/mass or joint cumulative probability. The general syntax is:

VoseDistributionProb({x}, {parameters}, cumulative, truncation)

where Distribution is replaced by the name of the distribution.

• {x} - a set of one or more values or cell references, on which the probability calculation is to be performed

• {parameters} - the parameters of the distribution

• Cumulative - an optional Boolean (TRUE/FALSE) parameter. If FALSE (default) the joint probability density for continuous distributions) or the joint probability mass for discrete distributions) is returned. If TRUE the joint cumulative probability is returned.

• Truncation - optional parameter that takes the form of either VoseXbounds(min,max) or VosePbounds(min,max), to truncate at specified x-values respectively p-values.

VoseDistributionProb10

Returns the Logarithm base 10 of the probability calculations described above. The general form is

VoseDistributionProb10({x}, {parameters}, cumulative, truncation)

where Distribution is replaced by the name of the distribution. This can be convenient, since the joint probability density/mass for a large set of values can quickly approach values too small for Excel to handle.

• {x} - a set of one or more values or cell references, on which the probability calculation is to be performed

• {parameters} - the parameters of the distribution

• Cumulative - an optional Boolean (TRUE/FALSE) parameter. If FALSE (default) the joint probability density for continuous distributions) or the joint probability mass for discrete distributions) is returned. If TRUE the joint cumulative probability is returned.

• Truncation - optional parameter that takes the form of either VoseXbounds(min,max) or VosePbounds(min,max), to truncate at specified x-values respectively p-values.

Examples

VoseBetaProb(0.3, 2, 5, FALSE) and VoseBetaProb(0.3, 2, 5) return the probability density of a Beta(2, 5) distribution at x = 0.3:

VoseBetaProb(0.3, 2, 5, 1) returns the cumulative probability of a Beta(2, 5) distribution at x = 0.3

VoseBetaProb({0.2,0.4,0.7}, 2, 5, FALSE) returns the joint probability density of a Beta(2, 5) distribution for the  values x = {0.2,0.4,0.7}:

VoseBetaProb({0.2,0.4,0.7}, 2, 5, TRUE) returns the joint cumulative probability of a Beta(2, 5) distribution for  the values x = {0.2,0.4,0.7}:

VoseBetaProb(A1:A6, 2, 5, TRUE) returns the joint cumulative probability of a Beta(2, 5) distribution for the values displayed in the spreadsheet range A1:A6

VoseBinomialProb(3, 12, 0.6, FALSE) and VoseBinomialProb(3, 12, 0.6) return the probability mass of a Binomial(12, 0.6) distribution at x = 3:

VosePoissonProb({3,4,7}, 5, 1) returns the cumulative probability of a Poisson(5) distribution at x = {3,4,7)

The image below demonstrates the principle of the function applied to a Beta distribution:

VoseBetaProb({0.3,0.6,0.8},3,4,0) calculates the joint density of the values {0.3,0.6,0.8} for a Beta(3,4) distribution. Result = 0.786579... which is the product of:

VoseBetaProb(0.3,3,4,0) = 1.8522...

VoseBetaProb(0.6,3,4,0) = 1.3824...

VoseBetaProb(0.8,3,4,0) = 0.3072...

VoseBetaProb({0.3,0.6,0.8},3,4,1) calculates the joint cumulative probability of the values {0.3,0.6,0.8} for a Beta(3,4) distribution. Result = 0.206310... which is the product of:

VoseBetaProb(0.3,3,4,1) = 0.25569...

VoseBetaProb(0.6,3,4,1) = 0.8208

VoseBetaProb(0.8,3,4,1) = 0.98304...