STAT 410
Homework #7
Fall 2011
(due Friday, October 21, by 3:00 p.m.)
0.
Warmup:
4.2.3
1.
Suppose
P
(
X
n
=
i
) =
6
3
+
+
n
i
n
,
for
i
= 1, 2, 3.
Find the limiting distribution of
X
n
.
2.
Let
X
n
have p.d.f.
f
n
(
x
) =
n
n
x
2
1
1
1
+
+
,
for
0 <
x
< 1,
zero elsewhere.
Find the limiting distribution of
X
n
.
3.
4.3.2
Let
Y
1
denote the minimum of a random sample of size
n
from a distribution that
has
p.d.f.
( )
( )
θ


=
x
e
x
f
,
θ
<
x
<
∞
,
zero elsewhere.
Let
Z
n
=
n
(
Y
1
–
θ
).
Investigate the limiting distribution of
Z
n
.
4.
4.3.3
Let
Y
n
denote the maximum
(
the last order statistic
) of a random sample of
size
n
from a distribution of the continuous type that has c.d.f.
F
(
x
)
and
p.d.f.
f
(
x
) = F
'
(
x
).
Find the limiting distribution of
Z
n
=
n
(
1 – F
(
Y
n
)
).
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View Full Document5.
a)
4.3.9
Let
X
be
( )
50
2
χ
.
Approximate
P
(
40 < X < 60
).
Hint:
We already know that
( )
( )
1
,
0
2
Y
N
D
n
n
n
→

.
b)
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 Fall '08
 Monrad
 Statistics, Probability, Probability theory, probability density function, m.g.f., corresponding order statistics.

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