Probability: Quiz II
May 25, 2004
1.
A filling station is supplied with gasoline once a week. Suppose its weekly volume of sales in
thousands of gallons is a random variable with probability density function
f
(
x
) =
‰
5(1

x
)
4
0
< x <
1
0
otherwise
How large must the capacity of the tank be so that the probability of the supply’s being exhausted in
a given week is 0.01? (5%)
Sol)
We want to find
c
such that
P
(
X > c
) = 1

F
(
c
) = 0
.
01. For 0
< x <
1, we have
F
(
x
) =
Z
x
0
5(1

t
)
4
dt
= 1

(1

x
)
5
Solving the following equation:
1

F
(
c
) = 1

[1

(1

c
)
5
] = 0
.
01
,
we get
c
= 1

0
.
01
1
/
5
≈
0
.
601. Hence, the tank capacity is required to be 601 gallons.
2.
The joint probability density function of X and Y is given by
f
(
x, y
) =
e

(
x
+
y
)
,
0
≤
x <
∞
,
0
≤
y <
∞
.
(a)
Find
P
(
X
≤
Y
). (5%)
Sol)
P
(
X
≤
Y
)
=
Z
∞
0
Z
∞
x
e

(
x
+
y
)
dydx
=
Z
∞
0
e

x
Z
∞
x
e

y
dydx
=
Z
∞
0
e

2
x
dx
=
1
2
(b)
Find
P
(
X
≤
a
). (5%)
Sol)
f
X
(
x
)
=
Z
∞
0
e

(
x
+
y
)
dy
=
e

x
,
x >
0
P
(
X
≤
a
)
=
Z
a
0
e

x
dx
= 1

e

a
,
a >
0
3.
Let
T
be the region bounded by the lines
y
= 0,
y
=
x
, and
x
= 1, as was shown in the following
figure.
y
= 0
x
= 1
y
=
x
x
y
Suppose that
f
is a function defined by
f
(
x, y
) =
cxy
for (
x, y
) in
T
, and that
f
(
x, y
) = 0 when (
x, y
)
is not in
T
.
1
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(a)
Find the value of
c
such that
f
(
x, y
) is a legitimate joint density function. (5%)
Sol)
1
=
Z
∞
0
Z
∞
0
f
(
x, y
)
dydx
=
c
Z
1
0
x
Z
x
0
ydydx
=
c
2
Z
1
0
x
3
dx
=
c
8
=
⇒
c
=
8
(b)
Find
P
(
Y >
1

X
). (5%)
Sol)
In the following figure, the darker area satisfies condition
Y >
1

X
.
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 Spring '11
 dat
 Probability theory, probability density function, 6.0 degrees, 8.0 degrees, 90.1 degrees

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