Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 123is the inverse function of g(x), or the solution for x in Equation (5.5) in terms ofy. Hence,F Y …y† ˆP…Y y† ˆP‰g…X† yŠ ˆP‰X g 1 …y†Š ˆ F X ‰g 1 …y†Š:…5:7†Equation (5.7) gives the relationship between the PDF of X and that of Y , ourdesired result.The relationship between the pdfs of X and Y are obtained by differentiatingboth sides of Equation (5.7) with respect to y. We have:f Y …y† ˆdF Y…y†dyˆ ddy fF X‰g 1 …y†Šg ˆ f X ‰g 1 …y†Š dg 1 …y†: …5:8†dyIt is clear that Equations (5.7) and (5.8) hold not only for the particulartransformation given by Equation (5.5) but for all continuous g(x) that are strictlymonotonic increasing functions of x, that is, g(x 2 )> g(x 1 )whenever x 2 >x 1 .Consider now a slightly different situation in which the transformation isgiven byY ˆ g…X† ˆ 2X ‡ 1: …5:9†Starting again with F Y (y) ˆ P(Y y), and reasoning as before, the regionY y in the range space R Y is now mapped into the region X > g 1 (y), asindicated in Figure 5.3. Hence, we have in this caseF Y …y† ˆP…Y y† ˆP‰X > g 1 …y†Šˆ 1 P‰X g 1 …y†Š ˆ 1 F X ‰g 1 …y†Š:…5:10†yy =–2x +1yxx = g –1 (y)= 1–y2Figure 5.3 Transformation defined by Equation (5.9)TLFeBOOK