Page 99 - Probability, Random Variables and Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES 9 1

The region corresponding to event A is expressed by x + y :I 2, which is shown in Fig. 3-3(a), that is,
the region below and including the straight line x + y = 2.
The region corresponding to event B is expressed by x2 + y2 < 2,, which is shown in Fig. 3-3(b), that
is, the region within the circle with its center at the origin and radius 2.
The region corresponding to event C is shown in Fig. 3-3(c), which is found by noting that

The region corresponding to event D is shown in Fig. 3-3(d), which is found by noting that

3.3. Verify Eqs. (3.7), (3.8a), and (3.8 b).

Since {X I Y I co) = S and by Eq. (1.22),
co,
P(X I a, Y I a) = FXy(a, a) = P(S) = 1
Next, as we know, from Eq. (2.8),
P(X = -Go)= P(Y = -co)=O
Since (X= - a,Y~y)c(X=-a) and (XIX,YI-a)c(Y=-co)
and by Eq. (1 .D), we have
P(X = -a, Y I y) = Fxy(-m, y) = 0
P(X I X, Y = - m) = FxY(x, -- m) = 0

3.4. Verify Eqs. (3.1 0) and (3.1 1).
Clearly (XSx,, Y IY)=(XSX,, Y ~y) u (x, <XIX,, Y ~y)
The two events on the right-hand side are disjoint; hence by Eq. (1 .D),
x,,
P(X r x,, Y I y) = P(X I Y I y) + P(x, < X I x,, Y 5 y)
or P(x1 <X5x2, YSy)= P(X IX,, Y ~y)- P(X IX,, Y ~y)
= Fxy(x2 9 Y) - Fxy(x1, Y)
Similarly,
(X I x, y 5 y,) = (X I x, Y I y,) u (X- I x, y, < Y < y,)
Again the two events on the right-hand side are disjoint, hence
P(X I x, Y I y,) = P(X I x, Y I y,) + P(X I x, y, < Y I y,)
or P(X 5 x, y, < Y I y,) = P(X 2 x, Y I y,) - P(X I x, Y S y,)
= Fx,(x, Y,) - FXY~
Y,)
3.5. Verify Eq. (3 .I 2).
Clearly
(x1 <XIx,, Y<y,)=(x, <XIx,, YIy,)v(x, <XIx,,y, < Y5y2)
The two events on the right-hand side are disjoint; hence
P(xl<XIx,, Ysy2)=P(xl <X<x2, Y<yl).f P(x1 <X<x,,y, < Ysy,)
Then using Eq. (3.1 O), we obtain