HW 3 Solution [100 points total]
Haoyang Liu
before lunar new year
[For graders: some will use decimals and other fractions. I try to give both when it is
not clear, but also check for yourselves.]
[For students: please email me if there is an error.]
problem 1
(a)
θ
(b)
θ
(c)
θ
(d)
θ

θ
2
(e) 3
θ
+ 4
(f) 9(
θ

θ
2
)
[grading scheme: 2 points each,12 points total]
problem 2
[putting numbers in a table is also fine]
(a) marginal for x:
Pr
(
x
= 1) = 0
.
2
, Pr
(
x
= 2) = 0
.
5
, Pr
(
x
= 3) = 0
.
3. marginal for y:
Pr
(
y
= 2) = 0
.
1
, Pr
(
y
= 3) = 0
.
3
, Pr
(
y
= 4) = 0
.
3
, Pr
(
y
= 5) = 0
.
3.
(b)
Pr
(
y
= 2

x
= 1) = 0
.
5
, Pr
(
y
= 3

x
= 1) = 0
.
5
Pr
(
y
= 3

x
= 2) = 0
.
4
, Pr
(
y
= 4

x
= 2) = 0
.
4
, Pr
(
y
= 5

x
= 2) = 0
.
2
(c)
E
(
y
) = 3
.
8 = 19
/
5
(d)
cov
(
x, y
) =
E
(
xy
)

ExEy
= 8
.
5

3
.
8
*
2
.
1 = 0
.
52
[grading scheme: 2 points each, 8 points total]
1
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Problem 3
[putting numbers in a table is also fine]
(a)
Pr
(
x
= 1) = 1
/
3
, Pr
(
x
= 2) = 1
/
3
, Pr
(
x
= 3) = 2
/
9
, Pr
(
x
= 4) = 1
/
9,
Pr
(
x
≤
1) =
1
/
3
, Pr
(
x
≤
2) = 2
/
3
, Pr
(
x
≤
3) = 8
/
9
, Pr
(
x
≤
4) = 1.
(b)
Pr
(
z
= 1) =
2
*
3
*
9

3
*
3
81
= 45
/
81 = 5
/
9
Pr
(
z
= 2) =
2
*
3
*
6

3
*
3
81
= 27
/
81 = 1
/
3
Pr
(
z
= 3) =
2
*
2
*
3

2
*
2
81
= 8
/
81
Pr
(
z
= 4) =
2
*
1
*
1

1
*
1
81
= 1
/
81
(c)
E
(
x
) =
E
(
y
) = 19
/
9
E
(
z
) = 127
/
81
(d)
Pr
(
x
= 1
, z
= 1) = 27
/
81
Pr
(
x
= 2
, z
= 1) = 9
/
81
, Pr
(
x
= 2
, z
= 2) = 18
/
81
Pr
(
x
= 3
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 Fall '08
 DRTON

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