FxX 1 () 112 F xxf xdxXX 12 ( ) ( )
X 2 22 (2.49)
The conditional moments of jointly distributed continuous random variables follow
straightforwardly from Equation (2.26) by use of Equation (2.47) and e.g. the jointly
distributed random variables X 1 , X 2 the conditional expected value XX 12 of X 1 given X 2 is
evaluated by:
XX 12 EXX x12 2 xf (^) XX 12
xx dx 2
(2.50)
2.10 Random Processes and Extremes
Random quantities may be “time variant” in the sense that they take on new realisations at
new trials or at new times. If the new realizations of the time variant random quantity occur at
discrete times and take on discrete realizations the random quantity is usually denoted a
random sequence. Well known examples hereof are series of throws of dices - more
engineering relevant examples are e.g. flooding events. If the realizations of the time variant
quantity occur continuously in time and take on continuous realizations the random quantity is
usually denoted a random process or stochastic process. Examples hereof are the wind
velocity, wave heights, snowfall and water levels.
In some cases random sequences and random processes may be represented in a given
problem context in terms of random variables e.g. for the modelling of the “point in time”
value of the intensity of the wind velocity, or the maximum (extreme) wind velocity during
one year. However, in many cases this is not possible and then it is necessary to model the
uncertain phenomena by a random process. In the following first a description of the Poisson
counting process is given and finally the continuous Normal or Gaussian processes are
described. It should be noted that numerous other types of random processes have been
suggested in the literature. In the lecture notes on the Basic Theory of Probability and
Statistics in Civil Engineering, (Faber, 2006) more information may be found.
The Poisson Counting Process
The most commonly applied family of discrete processes in structural reliability are the
Poisson processes. Due to the fact that Poisson processes have found applications in many
different types of engineering problems a large number of different variants of Poisson
processes has evolved. In general the process denoting the number of points in the
interval is called a simple Poisson process if it satisfies the following conditions:
Nt()
[0;t
The probability of one event in the interval [, is asymptotically proportional to the
length of the interval 2 t.
tt 2 t