Preliminaries#

This page is intended to provide the definitions of the basic concepts in probability and the notational conventions relevant to UQTestFuns. It is not intended to be an exhaustive treatment of the subject.

In the current version of UQTestFuns, we concern only with continuous input variables. Consequently, the discussion below is restricted to continuous random variables.

Note

Random variables are special mathematical objects1. A common convention is to write a random variable in upper case (\(X\), \(Y\), etc.) and the particular values of the random variable in lower case (\(x\), \(y\), etc.). But this convention is not always maintained here. For instance, some physical constants appearing in UQ test functions are denoted by capital letters (also by convention). Clearly, in this context, they are not random variables.

We would appeal to your wisdom in making the distinction between these notations.

Cumulative distribution function (CDF)#

A continuous random variable \(X\) distributed according to a (univariate) cumulative distribution function (CDF) \(F_X: \mathcal{D}_X \mapsto [0, 1]\) is written as:

\[ X \sim F_X (x). \]

The symbol \(\mathcal{D}_X\) above denotes the support of the random variable; that is, the set of all possible values of \(X\).

The CDF of \(X\) is by definition:

\[ F_X (x) \equiv \mathbb{P}[X \leq x] \]

The distribution of \(X\) is uniquely determined by its CDF.

Probability density function (PDF)#

The derivative of the cumulative distribution function, if exists, is called the (univariate) probability density function (PDF) of \(X\), denoted by \(f_X(x)\). Specifically, if exists, the PDF value at \(x\) is defined as follows:

\[ f_X(x) \equiv \frac{d F_X(x)}{dx} = \lim_{\Delta x \to 0} \frac{F_X(x + \Delta x) - F_X(x)}{\Delta x} \]

When this condition applies, then the PDF is also unique and we may also write:

\[ X \sim f_X(x). \]

It is important to realize that for a continuous random variable, the density value for a given \(x\) value is not the probability; it’s a density value. In fact, the probability of a \(X\) taking a particular value \(x\) is \(0\). You can, however, think of the quantity \(f_X(x) \, dx\) as a probability; it’s the probability of \(X\) having a value in an infinitesimal interval around \(x\).

With the PDF given we can define the support of a continuous random variable more precisely:

\[ \mathcal{D}_X \equiv \{ x \in \mathbb{R} | f_X (x) > 0 \}. \]

Using the PDF, we can write the CDF as follows:

\[ F_X (x) = \int_{\mathcal{D}^-_X}^x f_X (t) \, dt, \]

where \(\mathcal{D}^-_X\) denotes the lower bound of the support.

Inverse cumulative distribution function (ICDF)#

The inverse cumulative distribution function (ICDF) of a random variable \(X\) is the function \(F_X^{-1}: [0, 1] \mapsto \mathcal{D}_X\). In other words, the function takes a probability value \(p\) and returns the corresponding quantile value of the random variable such that \(F_X (x) = \mathbb{P}[X \leq x] = p\).

Some would refer to ICDF as the quantile function or percent point function (PPF).

Parametric distributions#

The distribution of a random variable may be one of the commonly known parametric distributions like the normal (Gaussian) distribution, the beta distribution, the Gumbel distribution, etc. As an example, if \(X\) is a normally distributed random variable, we write:

\[ X \sim \mathcal{N}(\mu, \sigma) \]

where \(\mu\) and \(\sigma\) are the parameters of the normal distributions (that is the mean and the standard deviation, respectively). Given the values of the parameters, a particular (parametric) distribution is completely and uniquely specified.

Multivariate random variables#

A multi-dimensional probabilistic input model is, in principle, a multivariate random variable; such a variable consists of individual univariate random variables grouped together under some dependency structure.

An \(M\)-variate random variable (or \(M\)-dimensional random vector) is denoted by \(\boldsymbol{X} = (X_1, \ldots, X_M )^T\) with a joint cumulative distribution function (CDF) \(F_{\boldsymbol{X}}: \mathcal{D}_{\boldsymbol{X}} \subseteq \mathbb{R}^M \mapsto [0, 1]\) and, if exists, a joint probability density function (PDF) \(f_{\boldsymbol{X}}: \mathcal{D}_{\boldsymbol{X}} \subseteq \mathbb{R}^M \mapsto \mathbb{R}_{\geq 0}\).

A common assumption (though in no way general) is the independence between all the component random variables in the random vector. Under independence, the CDF of an \(M\)-dimensional random vector \(\boldsymbol{X}\) can be written as follows:

\[ F_{\boldsymbol{X}} (x_1, \ldots, x_M) = \prod_{m = 1}^M F_{X_m} (x_m), \]

where \(F_{X_m}\) is the univariate CDF of a component random variable. If exists, the PDF can be written accordingly:

\[ f_{\boldsymbol{X}} (x_1, \ldots, x_M) = \prod_{m = 1}^M f_{X_m} (x_m), \]

where \(f_{X_m}\) is the univariate PDF of a component random variable.

1

it’s neither random (at least, not in completely unpredictable sense) nor a variable (it’s a function).