Introduction to Probability and Statistics for Engineers and Scientists

4.3Jointly Distributed Random Variables 97

Similarly, we can obtainP{Y =yj}by summingp(xi,yj) over all possible values ofxi,
that is,

P{Y=yj}=

∑

i

P{X=xi,Y=yj} (4.3.2)

=

∑

i

p(xi,yj)

Hence, specifyingthejointprobabilitymassfunctionalwaysdeterminestheindividualmass
functions. However, it should be noted that the reverse is not true. Namely, knowledge of
P{X=xi}andP{Y=yj}does not determine the value ofP{X=xi,Y=yj}.

EXAMPLE 4.3a Suppose that 3 batteries are randomly chosen from a group of 3 new, 4
used but still working, and 5 defective batteries. If we letXandY denote, respectively,
the number of new and used but still working batteries that are chosen, then the joint
probability mass function ofXandY,p(i,j)=P{X=i,Y=j}, is given by

p(0, 0)=

(

5

3

)/(

12

3

)

=10/220

p(0, 1)=

(

4

1

)(

5

2

)/(

12

3

)

=40/220

p(0, 2)=

(

4

2

)(

5

1

)/(

12

3

)

=30/220

p(0, 3)=

(

4

3

)/(

12

3

)

=4/220

p(1, 0)=

(

3

1

)(

5

2

)/(

12

3

)

=30/220

p(1, 1)=

(

3

1

)(

4

1

)(

5

1

)/(

12

3

)

=60/220

p(1, 2)=

(

3

1

)(

4

2

)/(

12

3

)

=18/220

p(2, 0)=

(

3

2

)(

5

1

)/(

12

3

)

=15/220

p(2, 1)=

(

3

2

)(

4

1

)/(

12

3

)

=12/220

p(3, 0)=

(

3

3

)/(

12

3

)

=1/220

These probabilities can most easily be expressed in tabular form as shown in Table 4.1.