Steven Weber
Dept. of ECE
Drexel University
ENGR 361: Statistical Analysis of Engineering Systems (Spring, 2010)
Midterm Exam Solutions
Instructions:
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You must answer three of them. Place a LARGE “X”
through the two problems you do not wish to be graded.
You have 50 minutes to complete the
exam. You are not allowed to use any books, notes, or calculators other than your 3”
×
5” note card.
Partial credit is given for answers that are partially correct. No credit is given for answers that are wrong
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Problem 1:
Problem 2:
Problem 3:
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www.ece.drexel.edu/weber
1
April 23, 2010
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Dept. of ECE
Drexel University
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Problem 1 (4 points)
Consider two dependent events
A,B
. Suppose
P
(
A
) =
p
and
P
(
B

A
) =
q
and
P
(
B

¯
A
) =
r
(here
¯
A
is the complement of
A
). Find an expression in terms of
p,q,r
for
P
(
A
∩
B

A
∪
B
)
.
(1)
Give a clear derivation of the expression with each step labeled (e.g., deﬁnition of conditional expectation, law of
total probability, etc.).
P
(
A
∩
B

A
∪
B
) =
P
((
A
∩
B
)
∩
(
A
∪
B
))
P
(
A
∪
B
)
(deﬁnition of conditional probability)
=
P
(
A
∩
B
)
P
(
A
∪
B
)
(
C
⊆
D
⇒
C
∩
D
=
C
)
=
P
(
B

A
)
P
(
A
)
P
(
A
∪
B
)
(deﬁnition of conditional probability)
=
P
(
B

A
)
P
(
A
)
P
(
A
∪
B

A
)
P
(
A
) +
P
(
A
∪
B

¯
A
)
P
(
¯
A
)
(law of total probability)
=
P
(
B

A
)
P
(
A
)
P
(
A
) +
P
(
B

¯
A
)
P
(
¯
A
)
=
qp
p
+
r
(1

p
)
(2)
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 Spring '04
 Eisenstein
 Conditional Probability, student id, Steven Weber

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