Fundamentals of Probability and Statistics for Engineers

142 Fundamentals of Probability and Statistics for EngineersFor independent X 1 and X 2 ,F Y …y† ˆˆZ 1 Z x2 y0Z 100f X1…x 1 †f X2…x 2 †dx 1 dx 2 ;F X1 …x 2 y†f X2…x 2 †dx 2 ; for y > 0;ˆ 0; elsewhere:…5:49†The pdf of Y is thus given by, on differentiating Equation (5.49) with respectto y,8 Z 1>< x 2 f X1…x 2 y†f X2…x 2 †dx 2 ; for y > 0;f Y …y† ˆ 0>:0; elsewhere;and, on substituting Equation (5.48) into Equation (5.50), it takes the form8 Z 1>< x 2 e x2y e x 21dx 2 ˆ ; for y > 0;2f Y …y† ˆ 0…1 ‡ y†>:0; elsewhere:Again, it is easy to check thatZ 10f Y …y†dy ˆ 1:…5:50†…5:51†Example 5.13. To show that it is elementary to obtain solutions to problemsdiscussed in this section when X 1 ,X 2 ,..., and X n are discrete, consider againY ˆ X 1 /X 2 given that X 1 and X 2 are discrete and their joint probability massfunction (jpmf) p X 1 X 2(x 1 ,x 2 ) is tabulated in Table 5.1. In this case, the pmf of Yis easily determined by assignment of probabilities p X 1 X 2(x 1 ,x 2 ) to the correspondingvalues of y ˆ x 1 /x 2 . Thus, we obtain:8>:0:12; for y ˆ 3:TLFeBOOK