Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 195To answer the first question, in part (a), we integrate f XY (x, y) over anappropriate region in the (x,y) plane satisfying y x. Since f XY (x, y) is aconstant over (0,0) (x,y) (1,1), this is the same as taking the ratio of thearea satisfying y x to the total area bounded by (0,0) (x,y) (1,1), whichis unity. As seen from Figure 7.4(a), we haveP…Y X† ˆshaded area A ˆ 12 :We proceed the same way in answering the second question, in part (b). It iseasy to see that the appropriate region for this part is the shaded area B, asshown in Figure 7.4(b). The desired probability is, after dividing area B into thetwo subregions as shown,P X Y X 1 ‡ ˆ shaded area B4ˆ14 3412 1 14 47ˆ :32We see from Example 7.2 that calculations of various probabilities of interestin this situation involve taking ratios of appropriate areas. If random variablesX and Y are independent and uniformly distributed over a region A, then theprobability of X and Y taking values in a subregion B is given byP‰…X; Y†in BŠ‡ˆarea of Barea of A :yy11AB—140 1x0 1…7:7† TLFeBOOKx(a)(b)Figure 7. 4 (a) Region A and (b) region B in the (x,y) plane in Example 7.2