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Similar to the univariate case, bivariate lognormal random variables have a PDF for x1, x2 > 0, in which where pln x and ulnx are the mean and standard deviation of log-transformed random variables, subscripts 1 and 2 indicate the random variables X1 and X2, respectively, and p12 = Corr(ln X1, lnX2) is the correlation […]

Instead of computing the exact value of Ф(z |Rx), several methods have been proposed to determine the bounds on the exact value of Ф(z |Rx). This section describes three such bounds. Bounds of Rackwitz. The scheme of Rackwitz (1978) is based on the decompo­sition of a positive correlation coefficient pij = XiXj, for i, j […]

Evaluation of the probability of multivariate normal random variables involves multidimensional integration as Accurate evaluation for Ф(г |Rx) is generally difficult and often is resolved by approximations. Bivariate normal probability. For abivariate case, Fortran computer programs for computing the lower left volume under the density surface, that is, Ф(а, b | p) = P(Z1 < […]

A bivariate normal distribution has a PDF defined as for k = 1 and 2. As can be seen, the two random variables having a bivariate normal PDF are, individually, normal random variables. It should be pointed out that given two normal marginal PDFs, one can construct a bivariate PDF that is not in the […]

Multivariate probability distributions are extensions of univariate probability distributions that jointly account for more than one random variable. Bivari­ate and trivariate distributions are special cases where two and three random variables, respectively, are involved. The fundamental basis of multivariate probability distributions is described in Sec. 2.3.2. In general, the availability of multivariate distribution models is […]

The normal distribution has been playing an important role in the development of statistical theories. This subsection briefly describes two distributions related to the functions of normal random variables. Figure 2.23 Shapes of standard beta probability density functions. (After Johnson and Kotz, 1972.)   X2 (chi-square) distribution. The sum of the squares of K independent […]

The beta distribution is used for describing random variables having both lower and upper bounds. Random variables in hydrosystems that are bounded on both limits include reservoir storage and groundwater table for unconfined aquifers. The nonstandard beta PDF is WI. —j—r(x — a)a 1(b — x)e 1 for a < x < b B(a, в)(b […]

Hydrosystems engineering reliability analysis often focuses on the statisti­cal characteristics of extreme events. For example, the design of flood-control structures may be concerned with the distribution of the largest events over the recorded period. On the other hand, the establishment of a drought- management plan or water-quality management scheme might be interested in the statistical […]

The gamma distribution is a versatile continuous distribution associated with a positive-valued random variable. The two-parameter gamma distribution hasa PDF defined as f g(x | а, в) = 1 (x/в)а—1 ex/fl for x > 0 (2.72) вГ(а) in which в > 0 and а > 0 are the parameters and Г(») is a gamma function […]

The lognormal distribution is a commonly used continuous distribution for posi­tively valued random variables. Lognormal random variables are closely related to normal random variables, by which a random variable X has a lognormal distribution if its logarithmic transform Y = ln(X) has a normal distribution with mean xln x and variance оЩx. From the central […]