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Random Variables and Probability Distributions 61

In closing this section, let us note that generalization to the case of many
random variables is again straightforward. The joint distribution function of n
random variables X 1 , X 2 ,..., X n , or X, is given, by Equation (3.19), as
F X x†ˆ P X 1 x 1 \ X 2 x 2 ... \ X n x n †: 3:31†

The corresponding joint density function, denoted by f ( x), is then
X

n
q F X x†
f x†ˆ ; 3:32†
X
qx 1 qx 2 ... qx n
if the indicated partial derivatives exist. Various properties possessed by these
functions can be readily inferred from those indicated for the two-random-
variable case.

3.4 CONDITIONAL DISTRIBUTION AND INDEPENDENCE

The important concepts of conditional probability and independence intro-
duced in Sections 2.2 and 2.4 play equally important roles in the context of
random variables. The conditional distribution function of a random variable X,
given that another random variable Y has taken a value y, is defined by

F XY xjy†ˆ P X xjY ˆ y†: 3:33†

Similarly, when random variable X is discrete, the definition of conditional mass
function of X given Y ˆ y is

p XY xjy†ˆ P X ˆ xjY ˆ y†: 3:34†

Using the definition of conditional probability given by Equation (2.24),
we have
P X ˆ x \ Y ˆ y†
p xjy†ˆ P X ˆ xjY ˆ y†ˆ ;
XY
P Y ˆ y†
or

p XY x; y†
p XY xjy†ˆ ; if p y† 6ˆ 0; 3:35†
Y
p y†
Y

TLFeBOOK