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.