Midterm 1 Solutions
1. Suppose that the time
X
(in hours) between successive commercials on a network
television station has the probability density function
f
X
(
x
) =
5
4

x
3
,
where 0
< x <
1
.
(a) Find the cumulative distribution function of
X
.
Solution.
F
X
(
x
) =
R
x
0
f
X
(
t
)
dt
=
R
x
0
(
5
4

t
3
)
dt
= (
5
4
t

1
4
t
4
)
ﬂ
ﬂ
x
0
=
5
4
x

1
4
x
4
.
(b) On average, how many hours of television uninterrupted by commercials would a
viewer of this station enjoy?
Solution.
μ
X
=
R
1
0
xf
X
(
x
)
dx
=
R
1
0
x
·
(
5
4

x
3
)
dx
= (
5
8
x
2

1
5
x
5
)
ﬂ
ﬂ
1
0
=
17
40
= 0
.
425
.
(c) What is the variance of the time in
minutes
between successive commercials on
this station?
Solution.
Var(
X
)
=
R
1
0
x
2
f
X
(
x
)
dx

μ
2
X
=
R
1
0
(
5
4
x
2

x
5
)
dx

(
17
40
)
2
= (
5
12
x
3

1
6
x
6
)
ﬂ
ﬂ
1
0

(
17
40
)
2
=
1
4

(
17
40
)
2
≈
0
.
069
.
If we let
Y
denote the time in minutes
between successive commercials, then
Y
= 60
X.
So Var(
Y
) = Var(60
X
) =
60
2
Var(
X
) = 249
.
75
.
2. A certain art museum has two paintings and two sculptures by Salvador Dal´
ı in its
collection. Because of considerations of space, the museum rarely displays all of its
Dal´
ı pieces simultaneously, but there is always at least one sculpture on display. Also,
the probability that all the pieces are displayed and the probability that exactly one
piece is displayed during a given week are both 0
.
1. Let
X
denote the number of Dal´
ı
paintings and
Y
the number of Dal´
ı sculptures on display during any given week. The
marginal probability distributions of
X
and
Y
are given below.
x
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 Fall '06
 Haskell
 Normal Distribution, Probability, Probability theory, probability density function, Cumulative distribution function

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