2.1.1. The probability distributions

There are several already-implemented statistical laws in Uranie, that can be called marginal laws as well, used to described the behaviour of a chosen input variable. They are usually characterised by two functions which are intrinsically connected: the PDF (probability density function) and CDF (cumulative distribution function). One can recap briefly the definition of these two functions for every random variable \(X : \Omega \rightarrow \rm I\!R\) :

  • PDF: if the random variable X has a density \(f_X\), where \(f_X\) is a non-negative Lebesgue-integrable function, then

\[P\left\{a\leq X \leq b \right\} = \int_{a}^{b} f_{X}(s)ds\]

  • CDF: the function \(F_{X} : {\rm I\!R} \rightarrow [0,1]\), given by

\[F_{X}(x)= \int_{-\infty}^{x}f_{X}(s)ds, \; x \in {\rm I\!R}\]

For some of the distributions discussed later on, the parameters provided to define them are not limiting the range of their PDF and CDF: these distributions are said to be infinite-based ones.
It is however possible to set boundaries in order to truncate the span of their possible values. One can indeed define an lower bound \(L\) and or an upper bound \(U\) so that the resulting distribution range is not infinite anymore but only in \([L, U]\). This truncation step affects both the PDF and CDF: once the boundaries are set, the CDF of these two values are computed to obtain \(P_{L}\) (the probability to be lower than the lower edge) and \(P_{U}\) (the probability to be lower than the upper edge). Two new functions, the truncated PDF \(f_{X}^{[L,U]}\) and the truncated CDF \(F_{X}^{[L,U]}\) are simply defined as

\[f_{X}^{[L,U]}(x)=\frac{f_{X}(x)}{P_{U}-P_{L}}, \;\; F_{X}^{[L,U]}(x) =\frac{F_{X}(x) - P_{L}}{P_{U}-P_{L}}.\]

These steps to produce a truncate distribution are represented in Figure 2.1 where the original distribution is shown on the left along with the definition of \(L\) (the blue shaded part) and \(U\) (the green shaded part). The right part of the plot is the resulting truncated PDF.

../../_images/TruncatedPrinciple.png

Figure 2.1 Principle of the truncated PDF generation (right-hand side) from the orginal one (left-hand side).

It is possible to combine different probability law, as a sum of weighted contributions, in order to create a new law. This approach, which is further discussed and illustrated in Composing law, leads to a new probability density function that would look like

\[f(x) = \sum_{j=1}^{N} \omega_j f_j(x)\;\; {\rm where} \;\; \forall j \in [1, N], \;\; \omega_j \in \mathbb{R}^{+}.\]

These distributions can be used to model the behaviour of variables, depending on chosen hypothesis, probability density function being used as a reference more oftenly by physicist, whereas statistical experts will generally use the cumulative distribution function [App13].

Table 2.1 gathers the list of implemented statistical laws, along with the list of parameters used to define them. For every possible law, a figure is displaying the PDF, CDF and inverse CDF for different sets of parameters (the equation of the corresponding PDF is reminded as well on every figure). The inverse CDF is basically the CDF whose x and y-axis are inverted (it is convenient to keep in mind what it looks like, as it will be used to produce design-of-experiments, later-on).

Table 2.1 List of Uranie classes representing the probability laws

Law

Class Uranie

Parameter 1

Parameter 2

Parameter 3

Parameter 4

Uniform

TUniformDistribution

Min

Max

Log-Uniform

TLogUniformDistribution

Min

Max

Triangular

TTriangularDistribution

Min

Max

Mode

Log-Triangular

TLogTriangularDistribution

Min

Max

Mode

Normal (Gauss)

TNormalDistribution

Mean (\(\mu\))

Sigma (\(\sigma\))

Log-Normal

TLogNormalDistribution

Mean (\(M\))

Error factor (\(E_f\))

Min

Trapezium

TTrapeziumDistribution

Min

Max

Low

Up

UniformByParts

TUniformByPartsDistribution

Min

Max

Median

Exponential

TExponentialDistribution

Rate (\(\lambda\))

Min

Cauchy

TCauchyDistribution

Scale (\(\gamma\))

Median

GumbelMax

TGumbelMaxDistribution

Mode (\(\mu\))

Scale (\(\beta\))

Weibull

TWeibullDistribution

Scale (\(\lambda\))

Shape (\(k\))

Min

Beta

TBetaDistribution

alpha (\(\alpha\))

beta (\(\beta\))

Min

Max

GenPareto

TGenParetoDistribution

Location (\(\mu\))

Scale (\(\sigma\))

Shape (\(\xi\))

Gamma

TGammaDistribution

Shape (\(\alpha\))

Scale (\(\beta\))

Location (\(\xi\))

InvGamma

TInvGammaDistribution

Shape (\(\alpha\))

Scale (\(\beta\))

Location (\(\xi\))

Student

TStudentDistribution

DoF (\(k\))

GeneralizedNormal

TGeneralizedNormalDistribution

Location (\(\mu\))

Scale (\(\alpha\))

Shape (\(\beta\))