ST 530 – February 28, 2005
Name:
Midterm #1
All problem parts have equal weight. In budgeting your time expect that
some problems will take longer than others.
1. Let
X
1
, . . . , X
n
be an i.i.d. sample from Exponential(1) distribution.
(a) Find
ES
2
n
, Var
S
2
n
. (Hint: When calculating
μ
4
think Γ functions).
(b) Find the density of
X
b
n/
2
c
:
n
.
(c) What is the asymptotic (approximate) distribution of
¯
X
n
and
X
b
n/
2
c
:
n
?
(d) Does
X
b
3
n/
4
c
:
n

X
b
n/
4
c
:
n
P
→
a
? If yes, ﬁnd the value of
a
.
(e) Is there a sequence
a
n
such that
X
n
:
n

a
n
D
→
Y
? If yes, ﬁnd it
and ﬁnd the distribution of
Y
. (Hint: Finding the c.d.f. of
Y
is
suﬃcient.)
Solution:
(a)
ES
2
n
= 1, Var
S
2
n
=
1
n
(
9

n

3
n

1
)
.
(b) Set
k
=
b
n/
2
c
,
f
X
(
k
)
(
x
) =
n
!
(
k

1)!(
n

k
)!
(1

e

x
)
k

1
e

(
n

k
+1)
x
I
(0
,
∞
)
(
x
)
.
(c)
¯
X
n
as.
∼
N
(1
,
1
n
). Since
ξ
1
/
2
= log 2 we have
X
b
n/
2
c
:
n
as.
∼
N
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 Summer '08
 Hannig,J
 Statistics, Probability distribution, Probability theory, probability density function, Yi, exponential family

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