Exam 1
(
Version B – yellow
)
(
Answers
)
1.
Suppose the momentgenerating function of a random variable X is
( 29
t
t
e
e
t
5
2
X
1
.
0
6
.
0
3
.
0
M
+
+
=
.
a)
(3)
Find the probability
P
(
X
≤
3
).
x
f
(
x
)
0
0.3
2
0.6
5
0.1
P
(
X
≤
3
)
=
f
(
0
) +
f
(
2
)
=
0.3 + 0.6
=
0.9
.
b)
(3)
Find the expected value of X,
E
(
X
).
M
X
'
(
t
)
=
t
t
e
e
5
2
5
.
0
2
.
1
+
.
E
(
Y
)
=
M
X
'
(
0
)
=
1.2 + 0.7
=
1.7
.
OR
x
f
(
x
)
x
⋅
f
(
x
)
0
0.3
0.0
2
0.6
1.2
5
0.1
0.5
1.7
E
(
X
)
=
∑
⋅
x
x
f
x
all
)
(
=
1.7
.
c)
(4)
Find the variance of X,
Var
(
X
).
M
X
''
(
t
)
=
t
t
e
e
5
2
5
.
2
4
.
2
+
.
E
(
X
2
)
=
M
X
''
(
0
)
=
2.4 + 2.5
=
4.9.
OR
x
f
(
x
)
x
2
⋅
f
(
x
)
0
0.3
0.0
2
0.6
2.4
5
0.1
2.5
4.9
=
E
(
X
2
).
Var
(
X
)
=
E
(
X
2
) – [
E
(
X
)
]
2
=
4.9 – 1.7
2
=
2.01
.
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View Full Document2.
Alex sets two alarm clocks each night to ensure that he does not sleep through
his 9:00 a.m. class.
His primary clock properly sounds its alarm on 80% of the
mornings, while his secondary clock rings its bell on only 70% of mornings.
Assume the clocks operate independently.
a)
(3)
What percent of the time does Alex’s twoclock strategy prevent him from
oversleeping?
That is, find the probability that at least one alarm would sound
on a given morning.
P( 1st OR 2nd ) = P( 1st ) + P( 2nd ) – P( 1st AND 2nd )
= 0.80 + 0.70 – 0.80
×
0.70 =
0.94
.
OR
P( at least one ) = 1 – P( none ) = 1 – 0.20
×
0.30 =
0.94
.
b)
(3)
Find the probability that only one alarm would sound on a given morning.
P( 1st only ) + P( 2nd only ) = 0.80
×
0.30 + 0.20
×
0.70 =
0.38
.
3.
(3)
Suppose an individual is randomly selected from the population of adult males
living in the United States.
Let
A
be the event that the selected individual is over
6 ft. in height, and let
B
be the event that the selected individual is a professional
basketball player.
Which do you think is larger, P(
A

B
)
or
P(
B

A
),
and
why
?
(
Circle one and
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 Spring '08
 STEPANOV
 Statistics, Poisson Distribution, Probability, Binomial distribution, Alex, Discrete probability distribution, professional basketball players

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