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How to Specify a Distribution Function
Monte Carlo Analysis
How to Specify a Distribution Function
By definition, “simple” random variables refer to statistical distributions that can easily be
generated on a computer. The context usually dictates which random variables are meant. For
example, the uniform distribution U([0, 1]) is simple, and so are the normal or user-defined
cumulative distributions in most circumstances. In other terms, the simple distributions are
elementary building blocks, and “complex” distributions can be defined as the result of complex
transformations of the other distributions. The simple random variables are often associated to
independent variables such as some process parameters or environmental conditions. In some
cases, the PDK models are derived from Principal Components Analysis, and the principal
components are simple Gaussian distributions.
The available distributions are grouped into two classes. The first class is associated to
parametric models. These models aim to describe probability distributions of a random variable
with the aid of a limited number of parameters Θ. The probability density of X can be expressed
as f X (x, Θ). The list of these distributions is given here:
• Uniform Distribution
• Gaussian or Normal Distribution
• Weighted Uniform and Gaussian Distributions
The second class represents distributions that are specified by a non-parametric procedure. A
typical example is given by the so-called weighted uniform or gaussian that is defined implicitly
by an accept/reject algorithm. The list of these distributions is given here:
• User-defined Distribution
• NOR/UNI Distributions
• Weighted Uniform and Gaussian Distributions
Note
Some of these distributions, such as the “weighted” UNIF/GAUSS, may seem unusual or
non-standard for many users. They represent the legacy of past experimentations in Eldo
and other third part tools. The implementation of those distributions are described.
Uniform Distribution
The uniform distribution defines equal probability over a given range for a continuous
distribution. The parameters are Θ = (L, U), with the constraint L < U. The probability density
function is expressed as:
Eldo® User's Manual, 15.3 431