This transition of the type of uncertainty has a significant importance because it facilitates that
the uncertainty is reduced by utilization of observations - updating.
2.9 Random Variables
The performance of an engineering system, facility or installation (in the following referred to
as system) may usually be modelled in mathematical physical terms in conjunction with
empirical relations. For a given set of model parameters the performance of the considered
system can be determined on the basis of this model. The basic random variables are defined
as the parameters that carry the entire uncertain input to the considered model.
The basic random variables must be able to represent all types of uncertainties that are
included in the analysis. The uncertainties, which must be considered are as previously
mentioned the physical uncertainty, the statistical uncertainty and the model uncertainty. The
physical uncertainties are typically uncertainties associated with the loading environment, the
geometry of the structure, the material properties and the repair qualities. The statistical
uncertainties arise due to incomplete statistical information e.g. due to a small number of
materials tests. Finally, the model uncertainties must be considered to account for the
uncertainty associated with the idealised mathematical descriptions used to approximate the
actual physical behaviour of the structure.
Modern methods of reliability and risk analysis allow for a very general representation of
these uncertainties ranging from non-stationary stochastic processes and fields to
time-invariant random variables, see e.g. Melchers (1987). In most cases it is sufficient to
model the uncertain quantities by random variables with given cumulative distribution
functions and distribution parameters estimated on the basis of statistical and/or subjective
information. Therefore the following is concerned with a basic description of the
characteristics of random variables.
Cumulative Distribution and Probability Density Functions
A random variable, which can take on any value, is called a continuous random variable. The
probability that such a random variable takes on a specific value is zero. The probability that a
continuous random variable, X, is less than or equal to a value, x, is given by the cumulative
distribution function:
Fx PX xX
(2.20)In general capital letters denote a random variable and small letters denote an outcome or
realization of a random variable. Figure 2.11 illustrates an example of a continuous
cumulative distribution function.