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5-1 TWO DISCRETE RANDOM VARIABLES 143

y
4.10 × 10 –5
4
f XY (x, y)
4.10 × 10 –5 1.84 × 10 –3
3
1.54 × 10 –5 1.38 × 10 –3 3.11 × 10 –2
2
Figure 5-1 Joint 2.56 × 10 –6 3.46 × 10 –4 1.56 × 10 –2 0.2333
1
probability distribution
of X and Y in Example 1.6 × 10 –7 2.88 × 10 –5 1.94 × 10 –3 5.83 × 10 –2 0.6561
0
5-1. 0 1 2 3 4
x
If X and Y are discrete random variables, the joint probability distribution of X and Y is a
description of the set of points (x, y) in the range of (X, Y) along with the probability of each point.
The joint probability distribution of two random variables is sometimes referred to as the bivari-
ate probability distribution or bivariate distribution of the random variables. One way to
describe the joint probability distribution of two discrete random variables is through a joint
probability mass function. Also, P(X x and Y y) is usually written as P(X x, Y y).

Definition
The joint probability mass function of the discrete random variables X and Y,
denoted as f XY (x, y), satisfies

(1) f 1x, y2 0
XY
(2) a a f XY 1x, y2 1

x y
(3) f XY 1x, y2 P1X x, Y y2 (5-1)

Subscripts are used to indicate the random variables in the bivariate probability distribution.
Just as the probability mass function of a single random variable X is assumed to be zero at all
values outside the range of X, so the joint probability mass function of X and Y is assumed to
be zero at values for which a probability is not specified.

EXAMPLE 5-2 Probabilities for each point in Fig. 5-1 are determined as follows. For example, P(X 2, Y 1)
is the probability that exactly two acceptable bits and exactly one suspect bit are received among
the four bits transferred. Let a, s, and u denote acceptable, suspect, and unacceptable bits, respec-
tively. By the assumption of independence,
P1aasu2 0.910.9210.08210.022 0.0013

The number of possible sequences consisting of two a’s, one s, and one u is shown in the CD
material for Chapter 2:

4!
12
2!1!1!

Therefore,

P1aasu2 1210.00132 0.0156