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EXAM P QUESTIONS OF THE WEEK
S. Broverman, 2005
Question 4  Week of August 15
A random variable
has a probability mass of 0.2 at
and a probability mass of 0.1 at
\\
œ
!
\œ"
\
. For all other values,
has the following density function:
, where
is a constant.
0Ð
B
Ñœ

!B
!
B
!B"
#B
" B 

\
H
(a) Find
TÐ
Þ&\"Ñ
(b) Find
TÐ\"
l\Þ&Ñ
(c) Find the mean of
.
\
The solution can be found below.
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View Full DocumentQuestion 4 Solution
(a)
TÐ
Þ&\"Ñœ
B.B œ Þ$(&Þ
'
Þ&
"
(b)
.
TÐ\"
l\Þ&Ñœ
TÐÞ&\"Ñ
TÐ\Þ&Ñ
From part (a), the numerator is
.
Þ$(&
Although we have not determined the value of
, we know that

TÐ\ Þ&Ñ œ " TÐ\ Ÿ Þ&Ñ œ " ÒTÐ\ œ !Ñ TÐ! \ Ÿ Þ&ÑÓ
œ " ÒÞ#
B .BÓ œ " Þ$#& œ Þ'(& Þ
'
!
Þ&
Then
.
TÐ\ "l\ Þ&Ñ œ
œ
œ Þ&&'
TÐÞ&\"Ñ
TÐ\Þ&Ñ
Þ'(&
Þ$(&
(c) In order for
to be a properly defined random variable, total probability over the probability
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 Spring '10
 Diaz
 Probability

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