Applied Statistics and Probability for Engineers

146 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

The calculation using the joint probability distribution can be used to determine E(X) even in

cases in which the marginal probability distribution of Xis not known. As practice, you can

use the joint probability distribution to verify that E(Y)  0.32 in Example 5-1.

Also,

Verify that the same result can be obtained from the joint probability distribution of Xand Y.

5-1.3 Conditional Probability Distributions

When two random variables are defined in a random experiment, knowledge of one can change

the probabilities that we associate with the values of the other. Recall that in Example 5-1, X

denotes the number of acceptable bits and Ydenotes the number of suspect bits received by a

receiver. Because only four bits are transmitted, if X4, Ymust equal 0. Using the notation for

conditional probabilities from Chapter 2, we can write this result as P(Y 0 X 4) 1. If

X 3, Ycan only equal 0 or 1. Consequently, the random variables Xand Ycan be considered

to be dependent. Knowledge of the value obtained for Xchanges the probabilities associated

with the values of Y.

Recall that the definition of conditional probability for events Aand Bis 

. This definition can be applied with the event Adefined to be X xand event
Bdefined to be Y y.

EXAMPLE 5-5 For Example 5-1, Xand Ydenote the number of acceptable and suspect bits received, respec-

tively. The remaining bits are unacceptable.

The probability that Y 1 given that X 3 is

Given that X 3, the only possible values for Yare 0 and 1. Notice that P(Y 0 X 3)

P(Y 1 X 3) 1. The values 0 and 1 for Yalong with the probabilities 0.200 and 0.800

define the conditional probability distribution of Ygiven that X3.

Example 5-5 illustrates that the conditional probabilities that Yygiven that X xcan be

thought of as a new probability distribution. The following definition generalizes these ideas.

fXY 1 3, 1 2 fX 132 0.2333 0.29160.800

P 1 Y 1 ƒX 32 P 1 X3, Y 12 P 1 X 32

fXY 1 3, 0 2 fX 132 0.05832 0.29160.200

P 1 Y 0 ƒX 32 P 1 X3, Y 02 P 1 X 32

P 1 A ̈B 2 P 1 A 2

P 1 BƒA 2

V 1 X 2 np 11 p 2  41 0.9 211 0.9 2 0.36

Definition

Given discrete random variables Xand Ywith joint probability mass function fXY(x,y)

the conditional probability mass functionof Ygiven Xxis

fY 0 x 1 y 2 fXY 1 x, y 2 fX 1 x 2 for fX 1 x 2

0 (5-4)

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