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Stat 210B Homework Assignment 1
(due January 25)
1. Show that if
EX
n
→
μ
and Var
X
n
→
0, then
X
n
P
→
μ
.
2. Show that if
∑
n
E

X
n

X

r
<
∞
, then
X
n
as
→
X
and
X
n
r
→
X
.
3. Suppose
X
n
is uniformly distributed on the set of points
{
1
/n,
2
/n, .
. . ,
1
}
. Show that
X
n
d
→
X
, where
X
is Un(0
,
1), where Un denotes the uniform distribution. Does
X
n
P
→
X
?
4. Given densities
p
n
and
q
n
with respect to some measure
μ
, deFne the likelihood ratio
L
n
(
x
)
as
L
n
(
x
) =
q
n
(
x
)
/p
n
(
x
) for
p
n
(
x
)
>
0,
L
n
(
x
) = 1 if
p
n
(
x
) =
q
n
(
x
) = 0 and
L
n
(
x
) =
∞
otherwise. Show that the likelihood ratio is a uniformly tight sequence.
5. Let
X
n
and
X
have densities
p
n
and
p
, respectively, with respect to a measure
μ
. Show
sup
B

P
(
X
n
∈
B
)

P
(
X
∈
B
)

=
1
2
Z

p
n

p

dμ,
where the supremum ranges over measurable sets
B
.
6. Let
X
1
, . . . , X
n
be a sample of size
n
from Be(
θ,
1), where
θ >
0. Let
¯
X
n
denote the sample
mean.
The methodofmoments estimate of
θ
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This note was uploaded on 01/25/2011 for the course STAT 201b taught by Professor Michaeljordan during the Fall '05 term at Berkeley.
 Fall '05
 MICHAELJORDAN
 Probability

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