Fundamentals of Probability and Statistics for Engineers

132 Fundamentals of Probability and Statistics for EngineersFor all y, Equation (5.25) yieldsf Y …y† ˆX2jˆ1f ‰gj 1 …y†Š dg j1 …y† dy a result identical to the solution for Example 5.4 [see Equation (5.18)].Ex ample 5 . 7. Problem: determine the pdf of Ydistributed according toˆ 1 12 1 ‡ y 2 ‡ 1 1…5:26†2 1 ‡ y 21ˆ…1 ‡ y 2 † ; 1 < y < 1;ˆ X 2 where X is normallyf X …x† ˆ 1…2† 1=2 e x2 =2 ; 1 < x < 1: …5:27†As shown in Figure 5.13, f Y (y) ˆ 0 for y < 0 since the transformationequation has no real roots in this range. For y 0, the two roots of2y ˆ x arex 1;2 ˆ g 11;2 …y† ˆy1=2 :yy = x 2yx 2 = –√yx 1 = √yxFigure 5. 13 Transformation y ˆ x 2TLFeBOOK