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Several continuous PDFs are used frequently in reliability analysis. They in­clude normal, lognormal, gamma, Weibull, and exponential distributions. Other distributions, such as beta and extremal distributions, also are used sometimes. The relations among the various continuous distributions considered in this chapter and others are shown in Fig. 2.15. 1.6.1 Normal (Gaussian) distribution The normal distribution […]

The Poisson distribution has the PMF as e — vV x px(x | v) =—;— for x = 0,1,2,… (2.53) x! where the parameter v > 0 represents the mean of a Poisson random variable. Unlike the binomial random variables, Poisson random variables have no upper bound. A recursive formula for calculating the Poisson PMF […]

The binomial distribution is applicable to random processes with only two types of outcomes. The state of components or subsystems in many hydrosystems can be classified as either functioning or failed, which is a typical example of a binary outcome. Consider an experiment involving a total of n independent trials with each trial having two […]

In the reliability analysis of hydrosystems engineering problems, several proba­bility distributions are used frequently. Based on the nature of the random vari­able, probability distributions are classified into discrete and continuous types. In this section, two discrete distributions, namely, the binomial distribution and the Poisson distribution, that are used commonly in hydrosystems reliability analysis, are described. […]

When a problem involves two dependent random variables, the degree of linear dependence between the two can be measured by the correlation coefficient pXyy, which is defined as Corr(X, Y) = px, y = Cov(X, Y )laTay (2.47) where Cov(X, Y) is the covariance between random variables X and Y, defined as Cov(X, Y) = […]

The asymmetry of the PDF of a random variable is measured by the skewness coefficient Yx, defined as E [(X — ,x)3]   _ _M3_ Yx = ,,1.5 ^2   (2.40)   a   The skewness coefficient is dimensionless and is related to the third-order central moment. The sign of the skewness coefficient indicates the […]

The spreading of a random variable over its range is measured by the variance, which is defined for the continuous case as / TO (X — fZx)2 fx(x) dx (2.36) -TO The variance is the second-order central moment. The positive square root of the variance is called the standard deviation ax, which is often used […]

In statistics, the term population is synonymous with the sample space, which describes the complete assemblage of all the values representative of a partic­ular random process. A sample is any subset of the population. Furthermore, parameters in a statistical model are quantities that are descriptive of the pop­ulation. In this book, Greek letters are used […]

The joint distribution and conditional distribution, analogous to the concepts of joint probability and conditional probability, are used for problems involving multiple random variables. For example, flood peak and flood volume often are considered simultaneously in the design and operation of a flood-control reservoir. In such cases, one would need to develop a joint PDF […]