Page 89 - Probability, Random Variables and Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES

D. Marginal Distribution Functions:
Now

since the condition y 5 oo is always satisfied. Then

Similarly, lim FXY~ Y) = FxY@, Y) = FY(Y) (3.14)
X+ 00
The cdf's FX(x) and F,(y), when obtained by Eqs. (3.13) and (3.14), are referred to as the marginal
cdf's of X and Y, respectively.

3.4 DISCRETE RANDOM VARIABLES-JOINT PROBABILITY MASS FUNCTIONS
A. Joint Probability Mass Functions :
Let (X, Y) be a discrete bivariate r.v., and let (X, Y) take on the values (xi, yj) for a certain
allowable set of integers i and j. Let
~xr(xi .Yj) = P(X = xi , Y = yj) (3.15)
The function pxy(xi, yj) is called the joint probability mass function (joint pmf) of (X, Y).

B. Properties of p&, , y,) :

where the summation is over the points (xi, yj) in the range space RA corresponding to the event A.
The joint cdf of a discrete bivariate r.v. (X, Y) is given by

C. Marginal Probability Mass Functions:
Suppose that for a fixed value X = xi, the r.v. Y can take on only the possible values yj (j = 1, 2,
. . . , n). Then

where the summation is taken over all possible pairs (xi, yj) with xi fixed. Similarly,

where the summation is taken over all possible pairs (xi, yj) with yj fixed. The pmf's pAxi) and pdyj),
when obtained by Eqs. (3.20) and (3.21), are referred to as the marginal pmf's of X and Y, respectively.

D. Independent Random Variables:
If X and Y are independent r.v.'s, then (Prob. 3.10)