Page 458 - Marine Structural Design

434 Part IY Structural Reliabiliy

A random variable is characterized by its probability density function p(x) and its cumulative
distribution function F,(x) = P[X I x] . The random variable is often described by its
statistical description, namely mean (or expected) value and variance (or standard deviation),
that are defined below
The n-th moment:
p,, = E[X"I n = 1,2,3, ... (24.1)

The n-fh central moment:
5, = E[(X - PI Y ] (24.2)

where,
p , = mean (or expected) value of X
s2 = ~ar[~] variance ofX
=
ox = & = standard deviation ofX

The mean value is the center of gravity of the probability density function. The standard
deviation is a measure of the dispersion around the mean value. The coefficient of variation
(CoV) is an uncertainty measure for the random variable X, which is defined by
The following non-dimensional values of the central moments may be defined.
JS,
Coefficient of variance: Co V = - (24.3)
PI

skewness: y = 53 (24.4)
dt2

Kurtosis: yz =& (24.5)
sf

24.2.3 Probabilistic Distributions
A random variable may be described by its cumulative distribution function. Some
distributions models are of special interests for stochastic and reliability analysis of marine
structures. These models are normal distribution, the lognormal distribution, the Rayleigh
distribution and the Weibull distribution, which are detailed below. Melchers (1999) also
defined other types of distribution function such as Poison, gamma, Beta, extreme value
distribution type I, II, III etc.
Normal (or Gaussian) Distribution
The probability density function and its cumulative distribution function for the normal
distribution are defined by

(24.6)