Applied Statistics and Probability for Engineers

5-1 TWO DISCRETE RANDOM VARIABLES 143

If Xand Yare discrete random variables, the joint probability distribution of Xand Yis 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 distributionor bivariate distributionof 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 xand Yy) is usually written as P(X x, Y y).

Subscripts are used to indicate the random variables in the bivariate probability distribution.

Just as the probability mass function of a single random variable Xis assumed to be zero at all

values outside the range of X, so the joint probability mass function of Xand Yis 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(X2, Y1)

is the probability that exactly two acceptable bits and exactly one suspect bit are received among

the four bits transferred. Let a,s, and udenote acceptable, suspect, and unacceptable bits, respec-

tively. By the assumption of independence,

The number of possible sequences consisting of two a’s, one s, and one uis shown in the CD

material for Chapter 2:

Therefore,

P 1 aasu 2  121 0.0013 2 0.0156

4!

2! 1! 1!

 12

P 1 aasu 2 0.9 1 0.9 21 0.08 21 0.02 2 0.0013

The joint probability mass function of the discrete random variables Xand Y,

denoted as fXY(x,y), satisfies

(1)

(2)

(3) fXY 1 x, y 2 P 1 Xx, Yy 2 (5-1)

a

x

(^) a
y
fXY 1 x, y 2  1
fXY 1 x, y 2  0
Definition
x
y
2.88 × 10 –51.94 × 10 –35.83 × 10 –20.6561
3.46 × 10 –41.56 × 10 –20.2333
1.38 × 10 –33.11 × 10 –2
1.84 × 10 –3
0 1 2 3 4
0
1
2
3
4
fXY(x, y)
1.6 × 10 –7
2.56 × 10 –6
1.54 × 10 –5
4.10 × 10 –5
4.10 × 10 –5
Figure 5-1 Joint
probability distribution
ofXandYin Example
5-1.
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