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Random Variables and Probability Distributions 51

F XY (x,y)
y

x

Figure 3.10 A joint probability distribution function of X and Y , F XY (x,y), when X and
Y are discrete
first quadrant. When both X and Y are continuous, F XY (x, y) becomes a smooth
surface with the same features. It is a staircase type in one direction and smooth in
the other if one of the random variables is discrete and the other continuous.
The joint probability distribution function of more than two random vari-
ables is defined in a similar fashion. Consider n random variables
X 1 , X 2 ,..., X n . Their JPDF is defined by

x 1 ; x 2 ; ... ; x n †ˆ P X 1 x 1 \ X 2 x 2 \ ... \ X n x n †: 3:19†
F X 1 X 2 ...X n

These random variables induce a probability distribution in an n-dimensional
Euclidean space. One can deduce immediately its properties in parallel to those
noted in Equations (3.17) and (3.18) for the two-random-variable case.
As we have mentioned previously, a finite number of random variables
X j , j ˆ 1, 2, ... n, may be regarded as the components of an n-dimensional
random vector X. The JPDF of X is identical to that given above but it can
be written in a more compact form, namely, F X ( x), where x is the vector, with

components x 1 , x 2 ,..., x n .

3.3.2 JOINT PROBABILITY MASS FUNCTION

The joint probability mass function (jpmf) is another, and more direct, charac-
terization of the joint behavior of two or more random variables when they are

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