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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 X is not known. As practice, you can
use the joint probability distribution to verify that E(Y) 0.32 in Example 5-1.
Also,

V1X 2 np11 p2 410.9211 0.92 0.36

Verify that the same result can be obtained from the joint probability distribution of X and 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 Y denotes the number of suspect bits received by a
receiver. Because only four bits are transmitted, if X 4, Y must 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, Y can only equal 0 or 1. Consequently, the random variables X and Y can be considered
to be dependent. Knowledge of the value obtained for X changes the probabilities associated
with the values of Y.
Recall that the definition of conditional probability for events A and B is P1B ƒ A2
P1A ¨ B2 P1A2 . This definition can be applied with the event A defined to be X x and event
B defined to be Y y.

EXAMPLE 5-5 For Example 5-1, X and Y denote the number of acceptable and suspect bits received, respec-
tively. The remaining bits are unacceptable.

P1Y 0 ƒ X 32 P1X 3, Y 02 P1X 32
f XY 13, 02 f X 132 0.05832 0.2916 0.200

The probability that Y 1 given that X 3 is

P1Y 1 ƒ X 32 P1X 3, Y 12 P1X 32
13, 12 f 132 0.2333 0.2916 0.800
f XY X
Given that X 3, the only possible values for Y are 0 and 1. Notice that P(Y 0 X 3)
P(Y 1 X 3) 1. The values 0 and 1 for Y along with the probabilities 0.200 and 0.800
define the conditional probability distribution of Y given that X 3.

Example 5-5 illustrates that the conditional probabilities that Y y given that X x can be
thought of as a new probability distribution. The following definition generalizes these ideas.

Definition
Given discrete random variables X and Y with joint probability mass function f XY (x, y)
the conditional probability mass function of Y given X x is
1y2 f 1x, y2 f 1x2 for f 1x2
0 (5-4)
f Y 0 x XY X X