Fundamentals of Probability and Statistics for Engineers

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 ,...,Xn,or X, is given, by Equation (3.19), as

The corresponding jo int den sity function, denoted by fX ( x), is then

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

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

Using the definition of conditional probability given by Equation (2.24),
we have

or

Random Variables and Probability D istributions 61

FX…x†ˆP…X 1 x 1 \X 2 x 2 ...\Xnxn†: … 3 : 31 †

fX…x†ˆ

qnFX…x†

qx 1 qx 2 ...qxn

; … 3 : 32 †

FXY…xjy†ˆP…XxjYˆy†: … 3 : 33 †

ˆ

pXY…xjy†ˆP…XˆxjYˆy†: … 3 : 34 †

pXY…xjy†ˆP…XˆxjYˆy†ˆ

P…Xˆx\Yˆy†

P…Yˆy†

;

pXY…xjy†ˆ

pXY…x;y†

pY…y†

;ifpY…y†6ˆ 0 ; … 3 : 35 †