PSTAT 120A Spring 2010 HW 9 Solutions
1. (Ross 6.40)
(a)
p
(
x, y
) is the joint PMF
p
(
x, y
) =
p
X,Y
(
x, y
) =
P
(
X
=
x, Y
=
y
) of discrete random variables
X
and
Y
.
The conditional PMF of
X
given
Y
can be obtained via
p
X

Y
(
x

y
) =
p
X,Y
(
x, y
)
p
Y
(
y
)
.
So we need the marginal of
Y
,
p
Y
(
y
) =
X
x
p
X,Y
(
x, y
)
,
p
Y
(1) =
p
X,Y
(1
,
1) +
p
X,Y
(2
,
1)
,
=
1
4
.
p
Y
(2) =
p
X,Y
(1
,
2) +
p
X,Y
(2
,
2)
,
=
3
4
.
Then the conditional PMF is
p
X

Y
(1

1) =
p
X,Y
(1
,
1)
p
Y
(1)
,
=
1
/
8
1
/
4
=
1
2
,
p
X

Y
(2

1) =
p
X,Y
(2
,
1)
p
Y
(1)
,
=
1
/
8
1
/
4
=
1
2
,
p
X

Y
(1

2) =
p
X,Y
(1
,
2)
p
Y
(2)
,
=
1
/
4
3
/
4
=
1
3
,
p
X

Y
(2

2) =
p
X,Y
(2
,
2)
p
Y
(2)
,
=
1
/
2
3
/
4
=
2
3
.
To be complete, we should also say
p
X

Y
(
x

y
) = 0
,
for
x /
∈ {
1
,
2
}
, y
∈ {
1
,
2
}
.
(b) For
X
and
Y
to be independent, it is necessary and sufficient that
p
X,Y
(
x, y
) =
p
X
(
x
)
p
Y
(
y
)
,
∀
(
x, y
)
∈
R
2
.
We find the marginal
p
X
(
x
) to check this criterion (we have
p
Y
(
y
) from part (a)).
p
X
(
x
) =
X
y
p
X,Y
(
x, y
)
,
p
X
(1) =
p
X,Y
(1
,
1) +
p
X,Y
(1
,
2)
,
=
3
8
.
p
X
(2) =
p
X,Y
(2
,
1) +
p
X,Y
(2
,
2)
,
=
5
8
.
1