Fundamentals of Probability and Statistics for Engineers

To answer the first question, in part (a), we integrate fX Y (x, y) over an
appropriate region in the (x,y) plane satisfying y x. Since fX Y (x, y) is a
constant over (0,0) (x,y) (1,1), this is the same as taking the ratio of the
area satisfying y x to the total area bounded by (0,0) (x,y) (1,1), which
is unity. As seen from Figure 7.4(a), we have

We proceed the same way in answering the second question, in part (b). It is
easy to see that the appropriate region for this part is the shaded area B, as
shown in Figure 7.4(b). The desired probability is, after dividing area B into the
two subregions as shown,

PX Y X

1

4

shaded area B

1

4

3

4

1

2

1

4

1

4

7

32

We see from Example 7.2 that calculations of various probabilities of interest
in this situation involve taking ratios of appropriate areas. If random variables
X and Y are independent and uniformly distributed over a region A, then the
probability of X and Y taking values in a subregion B is given by

(a) (b)

y

01 x

1

A

01 x

1

y

B

1

— 4

Figure 7. 4 (a) Region A and (b) region B in the (x,y) plane in Example 7.2

Some Important Continuous Distributions 195





 

P…YX†ˆshaded areaAˆ

1

2

:

‡



ˆ

ˆ



‡



ˆ :

P‰…X;Y†inBŠˆ

area ofB

area ofA

: … 7 : 7 †