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Spring 2008
Homework #1
(due Friday, January 25, 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)
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This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Monrad
 Probability

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