This preview shows pages 1–2. Sign up to view the full content.
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 we observe
X
1
,...,X
n
∼
N
(
μ
1
,σ
2
), and
Y
1
,...,Y
m
∼
N
(
μ
2
,σ
2
), and
X
1
,...,X
n
,
Y
1
,...,Y
m
are all independent of each other. Suppose we want to test whether
μ
1
=
μ
2
.
(1) If we known
σ
2
, then what is an appropriate test statistic, and what is the distribution of
this test statistic under the null hypothesis?
A: The test statistic is
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 03/30/2011 for the course STAT 100B taught by Professor Wu during the Winter '11 term at UCLA.
 Winter '11
 Wu

Click to edit the document details