ModelMaker Distributions

ModelMaker's Monte Carlo facility allows you to specify model parameters as random distributions. The distributions fall into two classes - Continuous and Discrete. The Probability Density Function (PDF) describes the probability that the value x lies in the range dx. For example, the probability that the value x lies in the range x < x1 is given by:

where p(x) is the Probability Density Function.

Continuous Distributions

A random variable is said to be continuous in a given range if it can assume any value in that range.

Normal Distribution

The Normal distribution generates random numbers according to a Gaussian distribution:

The user-supplied parameters are:

• Mean (μ)- the mean value of the distribution
• Standard Deviation (σ) - the standard deviation of the distribution about the mean

Triangular Distribution

The Triangular Distribution can be configured to produce both symmetrical and asymmetrical triangular distributions:

The user-supplied parameters are:

• Mode (b) - the apex of the triangular distribution
• Lower (a) - the lower limit of the distribution
• Upper (c) - the upper limit of the distribution

Uniform Distribution

The form of the Uniform distribution in the range a to b is:

The user-supplied parameters are:

• Bottom (a) - the lower limit of the distribution
• Top (b) - the upper limit of the distribution

Exponential Distribution

The Exponential distribution has the form:

There are no user-supplied parameters for this distribution.

Weibull Distribution

The Weibull distribution has the form:

The user-supplied parameters are:

• a - the parameter a in the expression above
• b - the parameter b in the expression above

Beta Distribution

The Beta distribution has the form:

where Γ(a), for example, refers to a value from the Gamma probability distribution for parameter a.
The user-supplied parameters are:

• a - the parameter a in the expression above
• b - the parameter b in the expression above

Gamma Distribution

The Gamma distribution of order a > 0 is defined by:

where, the scale factor b in the above expression is equal to 1.0 in ModelMaker 4.
The user-supplied parameter is:

• a - the mean of the distribution

Logistic Distribution

The Logistic distribution has the form:

This expression describes the distribution about a mean value of zero. The user-supplied parameters are:

• a - the mean value of the distribution
• mu (μ)- parameter controlling the width of the distribution

Pareto Distribution

The Pareto distribution has the form:

The user-supplied parameter is:

• a - the parameter a in the expression above

Extreme Value

The Extreme value distribution has the form:

The user-supplied parameters are:

• a - the parameter σ in the expression above
• b - the parameter μ in the expression above

Lognormal Distribution

The Lognormal distribution has the form:

Lognormal random numbers are the exponentials of Gaussian random numbers.
The user-supplied parameters are:

• Zeta - the parameter ζ in the expression above
• Sigma - the parameter σ in the expression above

Discrete Distributions