Fractiles of Random Variables
A fractile is the value of a random variable corresponding to a given probability of occurrence of values smaller than the fractile. It is an important concept used in many engineering and scientific applications. If a random variable is defined by a know
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Fractiles of Random Variables
A fractile is the value of a random variable corresponding to a given probability of occurrence of values smaller than the fractile. It is an important concept used in many engineering and scientific applications. If a random variable is defined by a known theoretical model then the fractile is simply the point at which the distribution function attains the specified probability. However, estimation of fractiles from limited sample data without having a theoretical model of the random variable is a more complicated task. Two different methods are commonly used: the classical coverage method and the prediction method. Operational techniques are provided for both methods and their comparison, taking into account the confidence level of the coverage method offered. In addition, the Bayessian approach to fractile estimation is explained, by way of updating prior data with newly obtained information. A review of fundamental procedures provides Annex 5.
9.1
Fractiles of Theoretical Models
One of the most important keywords in the theory of structural reliability is the term “fractile” of a random variable X (or of its probability distribution). In some publications and software products the term “quantile” [1, 2] is used, but more frequently the term fractile [3–5] is accepted (used also in this book). For a given probability p, the p-fractile xp denotes such a value of the random variable X, for which it holds that values of the variable X smaller than or equal to xp occur with the probability p. If Φ(x) is the distribution function of the random variable X, then it follows from Eq. (4.1) that the value Φ(xp) is equal to the probability p, thus the fractile xp can be defined as PðX xp Þ ¼ Φðxp Þ ¼ p
M. Holicky´, Introduction to Probability and Statistics for Engineers, DOI 10.1007/978-3-642-38300-7_9, © Springer-Verlag Berlin Heidelberg 2013
(9.1)
109
110
9 Fractiles of Random Variables
Distribution function F(u) 1.0
0.8
0.6
0.4
0.2
up
p 0.0 -3.0
-2.5
-2.0
-1.5
-1.0
-0.5
-1.0
-0.5
u
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Probability density j(u) 0.5
0.4
0.3
0.2
0.1
p up
0.0 -3.0
-2.5
-2.0
-1.5
u
Fig. 9.1 Definition of the fractile for a standardised random variable U
The same definition holds for a standardised random variable U (given by the transformation Eq. (4.23)), when in Eq. (9.1) U is substituted for X and up is substituted for xp. Figure 9.1 illustrates the definition given in Eq. (9.1). Fractiles up of standardised random variables U are commonly available in tables. Figure 9.1 illustrates the definition of the fractile described by Eq. (9.1) for a standardised random variable U; it shows the distribution function Φ(u), the probability density function φ(u), the probability p (equal to 0.05) and the fractile up
9.1 Fractiles of Theoretical Models
111
(equal to 1.645) for the distribution of a standardised variable U having the normal distribution. In general, the fractile xp of an original random variable
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