Fundamentals of Probability and Statistics for Engineers

146 Fundamentals of Probability and Statistics for EngineersR 2x 2yx 1 + x 2 = yx 1yFigure 5. 20 Region R 2 :x 1 ‡ x 2 yConsiderable importance is attached to the results expressed by Equations(5.54) and (5.55) because sums of random variables occur frequently in practicalsituations. By way of recognizing this fact, Equation (5.55) is repeated nowas Theorem 5.3.Theorem 5. 3. Let Y ˆ X 1 ‡ X 2 ,and let X 1 and X 2 be independent and continuousrandom variables. Then the pdf of Y is the convolution of the pdfsassociated with X 1 and X 2 ;that is,f Y …y† ˆZ 1f X1…y1x 2 †f X2…x 2 †dx 2 ˆZ 1f X2…y x 1 †f X1…x 1 †dx 1 : …5:56†1Repeated applications of this formula determine f Y (y) when Y is a sum ofany number of independent random variables.Ex ample 5. 16. Problem: determine f Y (y) of Y ˆ X 1 ‡ X 2 when X 1 and X 2 areindependent and identically distributed according tof X1…x 1 †ˆ ae ax 1; for x 1 0;…5:57†0; elsewhere;and similarly for X 2 .Answer: Equation (5.56) in this case leads toZ yf Y …y† ˆa 2 e a…y x2† e ax 2dx 2 ; y 0; …5:58†0TLFeBOOK