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STRUCTURAL RELIABILITY AND RISK ANALYSIS – 4 Year FILS

• it can be repeated arbitrarily often;
• the result of each performance depends on “chance” (that is, on influences which we cannot
control) and therefore can not be uniquely predicted.

The result of a single performance of the experiment is called the outcome of that trial. The
set of all possible outcomes of an experiment is called the sample space of the experiment and
will be denoted by S. Each outcome is called an element or point of S.

Experience shows that most random experiments exhibit statistical regularity or stability of
relative frequencies; that is, in several long sequences of such an experiment the
corresponding relative frequencies of an event are almost equal to probabilities. Since most
random experiments exhibit statistical regularity, one may assert that for any event E in such
an experiment there is a number P(E) such that the relative frequency of E in a great number
of performances of the experiment is approximately equal to P(E).

For this reason one postulates the existence of a number P(E) which is called probability of an
event E in that random experiment. Note that this number is not an absolute property of E but
refers to a certain sample space S, that is, to a certain random experiment.

1.4 Random variables

Roughly speaking, a random variable X (also called variate) is a function whose values are
real numbers and depend on chance (Kreyszig, 1979).

If one performs a random experiment and the event corresponding to a number a occurs, then
we say that in this trial the random variable X corresponding to that experiment has assumed
the value a. The corresponding probability is denoted by P(X=a).

Similarly, the probability of the event X assumes any value in the interval a<X<b is denoted
by P(a<X<b).

The probability of the event X≤ c (X assumes any value smaller than c or equal to c) is
denoted by P(X≤ c), and the probability of the event X>c (X assumes any value greater than c)
is denoted by P(X>c).

The last two events are mutually exclusive:

P(X≤c) + P(X>c) = P(-∞ < X < ∞) =1 (1.4)

The random variables are either discrete or continuous.

P( x  X  x  dx)  f ( x) dx - Definition of probability density function (PDF)
x
b

P( a  X  b)  f ( u) du (1.5)
a
Hence this probability equals the area under the curve of the density f(x) between x=a and
x=b, as shown in Figure 1.3.

UTCB, Technical University of Civil Engineering, Bucharest 10