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