University of Illinois
Fall 2009
ECE 313:
Final Exam
Tuesday, December 15, 2009, 8:00 a.m. — 11:00 a.m.
100 Materials Science and Engineering Building
1.
[60 points]
(3 points per answer)
Mark TRUE or FALSE for each statement below. No justiﬁcation is required, but to discourage
guessing, 3 points will be deducted for each incorrect answer (no penalty or gain for blank answers).
A net negative score will reduce your total exam score.
(a)
A
,
B
, and
C
are events such that 0
< P
(
A
)
<
1, 0
< P
(
B
)
<
1, and 0
< P
(
C
)
<
1.
TRUE
FALSE
±
±
P
(
A
∪
B
) =
P
(
A
c
∪
B
c
)

P
(
A
c
)

P
(
B
c
) + 1
.
±
±
P
(
AB
)
≥
P
(
A
) +
P
(
B
)

1
.
±
±
P
(
A
c

B
)
P
(
B
) +
P
(
A
c

B
c
)
P
(
B
c
) =
P
(
A
c
)
.
±
±
P
(
A
c

B
)
P
(
B
) +
P
(
A

B
)
P
(
B
) =
P
(
B
)
.
±
±
P
(
A
c

B
)
P
(
B
) +
P
(
A

B
c
)
P
(
B
c
) =
P
(
A
∪
B
)

P
(
AB
)
.
±
±
P
(
B

A
) =
P
(
A

B
)
P
(
A
)
/P
(
B
).
±
±
If
A
and
B
are
mutually exclusive
events, then they are
independent
events.
±
±
If
A
,
B
, and
C
are
independent
events, then
P
(
ABC
) =
P
(
A
)
P
(
B
)
P
(
C
).
±
±
If
P
(
ABC
) =
P
(
A
)
P
(
B
)
P
(
C
), then
A
,
B
, and
C
are
independent
events.
(b)
X
is a continuous random variable whose probability density function
f
X
(
u
) is an
even function
,
that is,
f
X
(
u
) =
f
X
(

u
) for
all
real numbers
u
. Let
F
X
(
u
) denote the cumulative probability
distribution function (CDF) of
X
, and assume that that
var
(
X
) = 4.
TRUE
FALSE
±
±
F
X
(
u
) =
F
X
(

u
) for all
u,
∞
< u <
∞
.
±
±
E
[
X
2
] = 4.
±
±
E
[

X

] = 2.
±
±
P
{
X
> u
}
=
F
X
(

u
) for all
u,
 ∞
< u <
∞
.
±
±
P
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 Spring '08
 Sarwate
 Probability distribution, Probability theory, probability density function, Tom, coin tosses

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