Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 231the PDF of Z can be shown to have the formF Z …z† ˆ1 expf exp‰…z u†Šg; 1 < z < 1 …7:105†where and u are again the two parameters of the distribution.It is seen that Type-I asymptotic distributions for maximum and minimumvalues are mirror images of each other. The mode of Z is u, and its mean,variance, and skewness coefficients are, respectively, 9m Z ˆ u >= 2 Z ˆ 2…7:106†6 2>; 1 ' 1:1396For probability calculations, values for probability distribution functionsF Y (y) and F Z (z) over various ranges of y and z are available in, for example,Microsoft Excel 2000 (see Appendix B).Ex ample 7. 9. Problem: the maximum daily gasoline demand Y during themonth of May at a given locality follows the Type-I asymptotic maximumvaluedistribution, with m Y ˆ 2and Y ˆ 1, measured in thousands of gallons.Determine (a) the probability that the demand will exceed 4000 gallons inany day during the month of May, and (b) the daily supply level that for 95%of the time will not be exceeded by demand in any given day.Answer: it follows from Equations (7.102) and (7.103) that parameters andu are determined fromFor part (a), the solution is ˆ p ˆ p ˆ 1:282;6 Y 60:577u ˆ m Y ˆ 2 0:5771:282 ˆ 1:55:P…Y > 4† ˆ1 F Y …4†ˆ 1 expf exp‰ 1:282…4 1:55†Šgˆ 1 0:958 ˆ 0:042:For part (b), we need to determine y such thatF Y …y† ˆP…Y y† ˆ0:95;TLFeBOOK