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Math 361, Problem set 7
Due 10/25/10
1. (1.9.6) Let the random variable
X
have
E
[
X
] =
μ
,
E
[(
X

μ
)
2
] =
σ
2
and
mgf
M
(
t
),

h < t < h
. Show that
E
±
X

μ
σ
²
= 0
,
E
"
³
X

μ
σ
´
2
#
= 1
and
E
±
exp
³
t
³
X

μ
σ
´´²
=
e

μt/σ
M
³
t
σ
´
,

hσ < t < hσ.
(Recall: exp(
x
) =
e
x
).
2. (1.9.7) Show that the moment generating function of the random variable
X
having pdf
f
(
x
) =
1
3
for

1
< x <
2, zero elsewhere is
M
(
t
) =
µ
e
2
t

e

t
3
t
t
6
= 0
1
t
= 0
3. (1.9.18) Find the moments of the distribution that has mfg
M
(
t
) = (1

t
)

3
,
t <
1.
Hint:
Find the MacLaurin’s series for
M
(
t
).
4. (1.9.23) Consider
k
continuoustype distributions with the following char
acteristics: pdf
f
i
(
x
), mean
μ
i
and variance
σ
2
i
,
i
= 1
,
2
,...,k
. If
c
i
≥
0,
i
= 1
,...,k
and
c
1
+
···
+
c
k
= 1, show that the mean and variance of
the distribution having pdf
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This note was uploaded on 10/26/2010 for the course MATHCS Math 316 taught by Professor Dr.paulhorn during the Fall '10 term at Emory.
 Fall '10
 Dr.PaulHorn

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