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STRUCTURAL RELIABILITY AND RISK ANALYSIS – 4 Year FILS

If a certain numerical value does not occur in the sample, its frequency is 0. If all the n values
of the sample are numerically equal, then this number has the frequency n and the relative
frequency is 1. Since these are the two extreme possible cases, one has:
- The relative frequency is at least equal to 0 and at most equal to 1;
- The sum of all relative frequencies in a sample equals 1.

If a sample consists of too many numerically different sample values, the process of grouping
may simplify the tabular and graphical representations, as follows (Kreyszig, 1979).

n 1 n 2 n j n m

x min x x x max x

A sample being given, one chooses an interval I that contains all the sample values. One
subdivides I into subintervals, which are called class intervals. The midpoints of these
subintervals are called class midpoints. The sample values in each such subinterval are said to
form a class.
The number of sample values in each such subinterval is called the corresponding class
frequency (absolute frequency - n j).
Division by the sample size n gives the relative class frequency (relative frequency -
n m f
N
f  j and f = 1). The normalized relative frequency is f  j . The normalization is
j
n  j j  x
 j 1
with respect to the x.
The relative frequency is called the frequency function of the grouped sample, and the
corresponding cumulative relative class frequency is called the distribution function of the
j
grouped sample (F j =  f ).
j
k 1
If one chooses few classes, the distribution of the grouped sample values becomes simpler but
a lot of information is lost, because the original sample values no longer appear explicitly.
When grouping the sample values the following rules should be obeyed (Kreyszig, 1979):
• All the class intervals should have the same length;
• The class intervals should be chosen so that the class midpoints correspond to simple number;
• If a sample value x j coincides with the common point of two class intervals, one takes it
into the class interval that extends from x j to the right.

1.3 Probability
Probability is an numerical measure of the chance or likelihood of occurrence of an event
relative to other events.
Letting n to move to infinite (n  ), frequencies moves to probabilities and consequently
f j probability. If x 0 then histogram of normalized relative frequencies become the
probability density function (PDF) and the histogram of cumulative relative frequencies
become the cumulative distribution function (CDF).

A random experiment or random observation is a process that has the following properties,
(Kreyszig, 1979):
• it is performed according to a set of rules that determines the performance completely;

UTCB, Technical University of Civil Engineering, Bucharest 9