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A random vector X = (X1, X2,…, XK)t has a multivariate normal distribution with a mean vector fj, x and covariance matrix Cx, denoted as X ~ N(px, Cx). The joint PDF of K normal random variables is given in Eq. (2.112). To gener­ate high-dimensional multivariate normal random variates with specified (j, x Drawdown recess […]

In preceding sections, discussions focused on generating univariate random variates. It is not uncommon for hydrosystems engineering problems to involve multiple random variables that are correlated and statistically dependent. For example, many data show that the peak discharge and volume of a runoff hy­drograph are positively correlated. To simulate systems involving correlated random variables, generated […]

The algorithms described in the preceding subsections are for some probability distributions commonly used in hydrosystems engineering and analysis. One might encounter other types of probability distributions in an analysis that are not described herein. There are several books that have been written for gen­erating univariate random numbers (Rubinstein, 1981; Dagpunar, 1988; Gould and Tobochnik, […]

The Poisson random variable is discrete, having a PMF fx(xi) = P (X = xi) given in Eq. (2.53). Dagpunar (1988) presented a simple algorithm and used the CDF-inverse method based on Eq. (6.7). When generating Poisson ran­dom variates, care should be taken so that e~v is not smaller than the ma­chine’s smallest positive real […]

The gamma distribution is used frequently in the statistical analysis of hydro­logic data. For example, Pearson type III and log-Pearson type III distributions used in the flood frequency analysis are members of the gamma distribution family. It is a very versatile distribution the PDF of which can take many forms (see Fig. 2.20). The PDF […]

Consider a random variable X having a lognormal distribution with a mean H-x and standard deviation ox, that is, X ~ LN(^x, ox). For a lognormal random variable X, its logarithmic transform Y = ln(X) leads to a normal distribution for Y. The PDF of X is given in Eq. (2.65). In the log-transformed space, […]

This section briefly outlines efficient algorithms for generating random variates for some probability distributions commonly used in hydrosystems engineering and analysis. 6.1.2 Normal distribution A normal random variable with a mean цx and standard deviation ox, denoted as X ~ N(p. x, ox), has a PDF given in Eq. (2.58). The relationship between X and […]

The variable transformation method generates a random variate of interest based on its known statistical relationship with other random variables the variates of which can be produced easily. For example, one is interested in generating chi-square random variates with n degrees of freedom. The CDF — inverse method is not appropriate in this case because […]

Consider a problem for which random variates are to be generated from a specified probability density function (PDF) fx(x). The basic idea of the TABLE 6.3 List of Distributions the Cumulative Distribution Function (CDF) Inverses of which Are Analytically Expressible Distribution Fx (x) = x = Fx fiu) Exponential 1 — exp(-вx), x > 0 […]

6.1.1 CDF-inverse method Let a random variable X have the cumulative distribution function (CDF) Fx(x). From Sec. 2.3.1, Fx(x) is a nondecreasing function with respect to the value of x, and 0 < Fx(x) < 1. Therefore, F-1(u) may be defined for any value of u between 0 and 1 as F-1(u) is the smallest […]