Fundamentals of Probability and Statistics for Engineers

134 Fundamentals of Probability and Statistics for EngineerswhereThe theorems stated in this section do not apply in this case to the portionsv 1 105V because infinite and noncountable number of rootsfor v 1 exist in these regions. However, we deduce immediately from Figure 5.14thatFor the middle portion, Equation (5.7) leads toNow,We thus haveg…V 1 †ˆ0; V 1 < 95;95g…V 1 †ˆV1; 95 V 1 105;10g…V 1 †ˆ1; V 1 > 105:P…V 2 ˆ 0† ˆP…V 1 95† ˆF V1 …95†Z 95ˆ f V1 …v 1 †dv 1 ˆ 1904 ;P…V 2 ˆ 1† ˆP…V 1 > 105† ˆ1 F V1 …105†ˆ 14 :F V2 …v 2 †ˆF V1 ‰g 1 …v 2 †Šˆ F V1 …10v 2 ‡ 95†; 0 < v 2 < 1:90F V1 …v 1 †ˆv1; 90 v 1 110:20F V2 …v 2 †ˆ 120 …10v 2 ‡ 95 90† ˆ2v 2 ‡ 1; 0 < v 2 < 1:4ThePDF, F V 2(v 2 ), is shown in Figure 5.15, an example of a mixed distribution.5.1.2 MOMENTSHaving developed methods of determining the probability distribution ofY ˆ g(X), it is a straightforward matter to calculate all the desired momentsTLFeBOOK