STAT 100B HW IV Due Friday
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
.
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 independent random errors, with E[
²
i
] = 0, Var[
²
i
] =
σ
2
. Suppose we want to estimate
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 Fall '08
 staff
 Normal Distribution, Probability, WI, STAT 100B HW, Suppose Yi

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