STAT 100B HW III solution
Problem 1:
Suppose
X
1
,...,X
n
∼
f
(
x
) independently. Let
μ
= E[
X
], and
σ
2
= Var[
X
]. Let
¯
X
=
1
n
n
X
i
=1
X
i
,
and
s
2
=
1
n

1
n
X
i
=1
(
X
i

¯
X
)
2
.
Prove E[
¯
X
] =
μ
, and E(
s
2
) =
σ
2
.
A: E[
¯
X
] = E[(
∑
n
i
=1
X
i
)
/n
] = (
∑
n
i
=1
E[
X
i
])
/n
=
nμ/n
=
μ
.
E[
s
2
] = E[
1
n

1
n
X
i
=1
(
X
i

¯
X
)
2
]
(1)
=
1
n

1
E[
n
X
i
=1
(
X
i

μ
)
2

n
(
¯
X

μ
)
2
]
(2)
=
1
n

1
[
n
X
i
=1
E(
X
i

μ
)
2

n
E(
¯
X

μ
)
2
]
(3)
=
1
n

1
(
nσ
2

n
(
σ
2
/n
)) =
σ
2
.
(4)
Problem 2:
Suppose
Y
i
=
x
i
β
true
+
²
i
,
i
= 1
,...,n
, where
x
i
are ﬁxed,
β
true
is an unknown
constant.
²
i
are random errors, with E[
²
i
] = 0, Var[
²
i
] =
σ
2
. Suppose we want to estimate
β
true
by
ˆ
β
=
∑
n
i
=1
w
i
Y
i
, where (
w
i
,i
= 1
,...,n
) may depend on (
x
i
,i
= 1
,...,n
). If we want
ˆ
β
to be unbiased
with minimum variance, then what should be the values of
w
i
,
i
= 1
,...,n
?
A: E[
ˆ
β
] =
∑
n
i
=1
w
i
E[
Y
i
] =
∑
n
i
=1
w
i
E[
x
i
β
true
+
²
i
] =
∑
n
i
=1
w
i
x
i
β
true
=
β
true
. So
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This note was uploaded on 01/27/2011 for the course STATS 100b taught by Professor Staff during the Fall '08 term at UCLA.
 Fall '08
 staff
 Probability

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