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STAT 410
Summer 2009
Homework #1
(due Friday, June 19, by 4:00 p.m.)
1.
Suppose a discrete random variable
X
has the following probability distribution:
P(
X =
k
) =
( 29
!
2
ln
k
k
,
k
= 1, 2, 3, … .
a)
Verify that this is a valid probability distribution.
b)
Find
μ
X
=
E
(
X
)
by finding the sum of the infinite series.
c)
Find the momentgenerating function of
X,
M
X
(
t
).
d)
Use
M
X
(
t
)
to find
μ
X
=
E
(
X
).
e)
Find
σ
X
2
=
Var
(
X
).
2.
Suppose a random variable
X
has the following probability density function:
≤
≤
=
otherwise
0
1
1
)
(
C
x
x
x
f
a)
What must the value of
C
be so that
f
(
x
)
is a probability density function?
b)
Find
P
(
X < 2
).
c)
Find
P
(
X < 3
).
d)
Find
μ
X
=
E
(
X
).
e)
Find
σ
X
2
=
Var
(
X
).
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View Full Document3.
An insurance policy reimburses a loss up to a benefit limit of 10.
The
policyholder’s loss,
Y, follows a distribution with density function:
f
(
y
)
=
otherwise
0
1
if
2
3
y
y
a)
What is the expected value and the variance of the policyholder’s loss?
b)
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 Spring '08
 STEPANOV
 Statistics, Probability

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