ModelMaker Probability 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 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 Discrete random variables may only take on distinct values. For the distributions that follow, the equations describe the probability for a particular whole number k, denoted P(k). Binomial Distribution The binomial distribution has the form: The user-supplied parameters are: Value - the value p in the expression above b - the parameter n in the expression above Poisson Distribution The Poisson distribution has the form: The user-supplied parameters are: Mean - μ in the expression above) Geometric Distribution The Geometric distribution has the form: The user-supplied parameters are: Parameter a Probability (p) - the probability that the event occurs
ModelKinetix is a trade name of FamilyGenetix, managed by AP Benson. © FamilyGenetix 2001-2003