Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 233where u is the scale factor and the value of u describes the characteristics ofa river; it varies from 1.5 for violent rivers to 10 for stable or mild rivers.In closing, let us remark again that the Type-I maximum-value distributionis valid for initial distributions of such practical importance as normal, lognormal,and gamma distributions. It thus has wide applicability and is sometimessimply called the extreme value distribution.7.6.2 TYPE-II ASYMPTOTIC DISTRIBUTIONS OF EXTREMEVALUESThe Type-II asymptotic distribution of maximum values arises as the limitingdistribution of Y n as n !1from an initial distribution of the Pareto type, thatis, the PDF F X (x) of each X j is limited on the left at zero and its right tail isunbounded and approaches one according toF X …x† ˆ1 ax k ; a; k > 0; x 0: …7:111†For this class, the asymptotic distribution of Y n , F Y (y), as n !1takes theformF Y …y† ˆexpy k; v; k > 0; y 0: …7:112†vLet us note that, with F X (x) given by Equation (7.111), each X j has momentsonly up to order r, where r is the largest integer less than k. If k > 1, the mean ofY ism Y ˆ v 11; …7:113†kand, if k > 2, the variance has the form 2 Y ˆ 2v2 1k211k: …7:114†The derivation of F Y (y) given by Equation (7.112) follows in broad outlinethat given for the Type-I maximum-value asymptotic distribution and will notbe presented here. It has been used as a model in meteorology and hydrology(Gumbel, 1958).A close relationship exists between the Type-I and Type-II asymptoticmaximum-value distributions. Let Y I and Y II denote, respectively, these randomTLFeBOOK