Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 129yy 2yy =g(x)y 1x 1 = g 1 (y)x 2 = g 2 (y)x 3 = g 3 (y)–1 –1 –1xFigure 5. 10 An example of nonmonotonic function y ˆ g(x)In the examples given above, it is easy to verify that all density functionsobtained satisfy the required properties.Let us now turn our attention to a more general case where functionY ˆg(X) is not necessarily strictly monotonic. Two examples are given inFigures 5.10 and 5.11. In Figure 5.10, the monotonic property of the transformationholds for y < y 1 , and y>y 2 , and Equation (5.12) can be used todetermine the pdf of Y in these intervals of y. For y 1 y y 2 , however, wemust start from the beginning and consider F Y (y) ˆP(Y y). The regiondefined by Y y in the range space R Y covers the heavier portions of thefunction y ˆg(x), as shown in Figure 5.10. Thus:F Y …y† ˆP…Y y† ˆP‰X g1 11…y†Š ‡ P‰g2 …y† < X g 3 1 …y†Šˆ P‰X g 1111…y†Š ‡ P‰X g3 …y†Š P‰X g2 …y†Šˆ F X ‰g 11 …y†Š ‡ F X‰g 13 …y†Š F X‰g 12 …y†Š; y 1 y y 2 ;…5:21†where x 1 ˆ g1 1y),x 2 ˆ g2 1y),andx 3 ˆ g3 1y)are roots for x of functiony ˆ g( x) in terms of y.As before, the relationship between the pdfs of X and Y is found by differentiatingEquation (5.21) with respect to y. It is given byf Y …y† ˆf X ‰g 111dg1 …y†Š …y†dy‡ f X ‰g 131dg3 …y†Š …y†dyf X ‰g 121dg2 …y†Š …y† ; y 1 y y 2 :dy…5:22†TLFeBOOK