STAT 410
Fall 2008
Homework #1
(due Friday, September 5, by 3:00 p.m.)
1.
Consider a continuous random variable
X
with probability density function
f
X
(
x
)
=
°
±
°
²
³
<
<
o.w.
0
1
0
3
2
x
x
Find the momentgenerating function of X, M
X
(
t
)
.
2.
Suppose a discrete random variable
X
has the following probability distribution:
P(
X
=
k
) =
(
)
!
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
)
.
3.
Suppose a random variable
X
has the following probability density function:
°
±
°
²
³
≤
≤
⋅
=

otherwise
0
1
0
)
(
x
C
x
f
x
e
a)
What must the value of
C
be so that
f
(
x
)
is a probability density function?
b)
Find the cumulative distribution function
F
(
x
)
= P
(
X
≤
x
)
.
c)
Find the median of the probability distribution of
X
.
d)
Find
μ
X
=
E
(
X
)
.
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4.
Suppose a random variable
X
has the following probability density function:
°
±
°
²
³
≤
≤
=
otherwise
0
1
1
)
(
C
x
x
x
f
a)
What must the value of
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 Fall '08
 AlexeiStepanov
 Normal Distribution, Probability, Probability distribution, Probability theory, probability density function

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