Page 1 of 6
Exam 1
(
Version B – yellow
)
(
Answers
)
1.
Consider two continuous random variables X and Y with joint p.d.f.
f
X,
Y
(
x
,
y
) =
±
²
³
<
<
<
<
otherwise
0
0
,
0
81
2
2
K
K
y
x
y
x
a) (5) Find the value of
K
so that
f
X,
Y
(
x
,
y
) is a valid joint p.d.f.
1 =
´ ´
K K
dy
dx
y
x
0 0
2
81
2
=
243
5
K
.
µ
K
=
3
.
b) (5) Find P
(
X > 3
Y
).
P
(
X > 3
Y
) =
´
´
¶
¶
¶
·
¸
¹
¹
¹
º
»
3
0
3
0
2
81
2
dx
dy
y
x
x
=
´
3
0
4
729
1
dx
x
=
15
1
.
OR
P
(
X > 3
Y
) =
´
´
¶
¶
¶
·
¸
¹
¹
¹
º
»
1
0
3
3
2
81
2
dy
dx
y
x
y
= … =
15
1
.
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2.
(7) Determine the mean and variance of a random variable with the following
momentgenerating function:
M
X
(
t
) =
e
t
(
7 + 8
t
)
=
exp
[
t
(
7 + 8
t
)
].
M
X
(
t
) =
exp
[
t
(
7 + 8
t
)
] =
exp
(
7
t
+ 8
t
2
).
Normal distribution:
M
X
(
t
) =
2
2
2
±
t
t
e
+
.
X has a Normal distribution with
μ
= 7 and
σ
2
/
2
= 8.
E
(
X
) =
μ
=
7
,
Var
(
X
) =
σ
2
=
16
.
OR
M
X
'
(
t
) = (
7 + 16
t
)
exp
(
7
t
+ 8
t
2
).
E
(
X
) = M
X
'
(
0
) =
7
.
M
X
"
(
t
) = 16
exp
(
7
t
+ 8
t
2
) + (
7 + 16
t
)
2
exp
(
7
t
+ 8
t
2
).
E
(
X
2
) = M
X
"
(
0
) = 65.
Var
(
X
) = 65 – 7
2
=
16
.
3.
(6) A machine fastens plastic screwon caps onto containers of motor oil. If the machine
applies more torque than the cap can withstand, the cap will break. Both the torque
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 Spring '08
 AlexeiStepanov
 Normal Distribution, Probability, Variance, Probability theory, Alex, probability density function

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