Fundamentals of Probability and Statistics for Engineers

230 Fundamentals of Probability and Statistics for EngineersThe mean and variance associated with the Type-I maximum-value distributioncan be obtained through integration using Equation (7.90). We have notedthat u is the mode of the distribution, that is, the value of y at which f Y (y) ismaximum. The mean of Y ism Y ˆ u ‡ ;…7:102†where ' 0:577 is Euler’s constant; and the variance is given by 2 Y ˆ 26 2 :…7:103†It is seen from the above that u and are, respectively, the location and scaleparameters of the distribution. It is interesting to note that the skewnesscoefficient, defined by Equation (4.11), in this case is 1 ' 1:1396;which is independent of and u. This result indicates that the Type-Imaximum-value distribution has a fixed shape with a dominant tail to the right.A typical shape for f Y (y) is shown in Figure 7.14.The Type-I asymptotic distribution for minimum values is the limitingdistribution of Z n in Equation (7.91) as n !1 from an initial distributionF X (x) of which the left tail is unbounded and is of exponential type as it decreasesto zero on the left. An example of F X (x) that belongs to this class is the normaldistribution.The distribution of Z n as n !1can be derived by means of proceduresgiven above for Y n through use of a symmetrical argument. Without givingdetails, if we letlimn!1 Z n ˆ Z; …7:104†f Y (y )yFigure 7.14Typical plot of a Type-I maximum-value distributionTLFeBOOK