Page 90 - Probability, Random Variables and Random Processes

82 MULTIPLE RANDOM VARIABLES [CHAP 3

3.5 CONTINUOUS RANDOM VARIABLES-JOINT PROBABILITY DENSITY
FUNCTIONS
A. Joint Probability Density Functions:

Let (X, Y) be a continuous bivariate r.v. with cdf FXdx, y) and let

The function fxy(x, y) is called the joint probability density function (joint pdf) of (X, Y). By
integrating Eq. (3.23), we have

B. Properties of f,(x, y):

3. f,Ax, y) is continuous for all values of x or y except possibly a finite set.

Since P(X = a) = 0 = P(Y = c) [by Eq. (2.19)], it follows that

C. Marginal Probability Density Functions:
By Eq. (3.13)'

Hence

or

Similarly,

The pdf's fdx) and fdy), when obtained by Eqs. (3.30) and (3.31), are referred to as the marginal pdf's
of X and Y, respectively.