Page 88 - Probability, Random Variables and Random Processes

80 MULTIPLE RANDOM VARIABLES [CHAP 3

3.3 JOINT DISTRIBUTION FUNCTIONS
A. Definition:
The joint cumulative distribution function (or joint cdf) of X and Y, denoted by FXy(x, y), is the
function defined by
Fxy(x, y) = P(X I Y _< y) (3.1)
x,
in
The event (X 5 x, Y I Eq. (3.1) is equivalent to the event A n B, where A and B are events of S
y)
defined by
A = {c E S; X(c) _< x) and B = (5 E S; Y(() I y) (3-2)
and 44 = Fx(x) P(B) = FY(Y)
Thus, FXY~ = P(A n B) (3.3)
Y)
If, for particular values of x and y, A and B were independent events of S, then by Eq. (l.46),
FXYk y) = P(A n B) = WWB) = Fx(x)FAy)

B. Independent Random Variables:
Two r.v.'s X and Y will be called independent if
Fxy(x9 Y) = FX(X)FY(Y)
for every value of x and y.

C. Properties of F&, y):
The joint cdf of two r.v.'s has many properties analogous to those of the cdf of a single r.v.

0 I Fxy(x, y) < 1 (3.5)
If x1 I x, , and y1 I y2 , then
Y
FxY~, 5 Fxy(x2 , Y 1) 5 Fxy(x2 , Y2) (3.6~)
1)
YJ < Fxy(x1, Y2) l Fxy(x2 9 Y2)
FXY~ (3.6b)
lirn FXy(x, y) = Fxy(w, GO) = 1 (3.7)
x-'m
Y+W
y)
lim FXy(x, y) = Fxy(- a, = 0 (3.8~)
X+ - W
lim FXy(x, y) = FXy(x, - co) = 0 (3.8 b)
y---co
lim FXY(X, Y) = Fxy(af, Y) = Fxy(a, Y) (3.9~)
x+a+
lim FXy(x, y) = FXy(x, b +) = FXy(x, b) (3.9b)
y-+b+
P(x1 < X I x2, Y 5 Y) = Fxy(x2, YbFxy(x1, Y) (3.1 0)
Y2)
P(X 5 x, YI < Y 5 Y2) = FXY~ - FXY(X, YI) (3.1 1)
If x, I and y, I y,, then
x,
Fxy(x2 7 Y2) - FxY(% Y2) - FXY(X~ Yd + FXY~ 2 0 (3.1 2)
Yl)
3
y,)
Note that the left-hand side of Eq. (3.12) is equal to P(xl < X 5 x2, y, < Y I (Prob. 3.5).