NAME:
MAE 108 – Probability and Statistical Methods for Engineers  Winter 2010
Midterm # 2
 February 22, 2010
50 minutes, open book, open notes, calculator allowed, no cell phones.
Write your answers directly on the exam. 40 points total.
(1) (10 points)
Consider a random variable
X
. The PDF for
X
is given by
f
X
(
x
) =
ce

2
x
,
0
< x <
∞
and zero otherwise.
(a) Find the value of
c
Solution:
R
∞
0
f
X
(
x
)
dx
= 1 so
c
= 2.
(b) Calculate
P
(
X >
2).
Solution:
P
(
X >
2) =
R
∞
2
2
e

2
x
dx
= [

e

2
x
]
∞
2
=
e

4
≈
1
.
83%
(c) Calculate the mean
E
(
X
)
Solution:
E
(
X
) =
R
∞
0
2
xe

2
x
dx
=
1
2
R
∞
0
ue

u
du
=
1
2
by integration by parts
(d) Calculate the standard deviation
σ
X
.
Solution:
E
(
X
2
) =
R
∞
0
2
x
2
e

2
x
dx
=
1
4
R
∞
0
u
2
e

u
du
=
1
2
by integration by parts so
σ
X
=
p
E
(
X
2
)

E
(
X
)
2
=
q
1
2

1
4
=
1
2
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(2) (10 points)
The SAT mathematics test scores across the population of high school seniors follow a normal
distribution with mean 500 and standard deviation 100. If five seniors are randomly chosen, find
the probability that:
(a) All of them scored below 600.
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 Winter '10
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 Probability, Probability theory, #, Poisson process, 3.67%

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