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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Civil and Environmental Engineering
1.017/1.010 Computing and Data Analysis for Environmental Applications/
Uncertainty in Engineering
Quiz 2 (Solutions provided at the end of each problem)
Tuesday, November 4, 2003
Problem 1 ( 15 points)
Sometimes we may want to generate two correlated random variables for stochastic simulation
applications.
For example, the duration and intensity of rain storm may be highly uncertain but
positively correlated.
One option for generating correlated variables
x
and
y
is to obtain
y
from:
y
=
ax
+
b
ε
where
x
and
are independent random variables with means equal to 0.0 and variances equal to 1.0
and
a
and
b
are specified constants.
a. (5 points)
What are the mean and variance of
y
? (expressed as functions of
a
and
b
):
Solution:
E
[
y
]
= E
[
ax+b
]
= aE
[
x
]
+bE
[
]
= 0
Var
[
y
]
= Var
[
ax+b
]
= a
2
Var
[
x
]
+ b
2
Var
[
]
= a
2
+ b
2
b. (10 points)
Select
a
and
b
so that
y
has a variance of 2 and the correlation between
x
and
y
is
0.5.
Show all relevant calculations.
Solution:
a
2
+ b
2
=
2
Correl
(
x,y
)
=
0.5
= Cov
(
x,y
)
/
(
Std
[
x
]
Std
[
y
])
Cov
(
x,y
)
= E
[(
x
x
)(
y
y
)]
y
y = a
(
x
x
)
+b
(

)
so:
Cov
(
x,y
)
= E
[
a
(
x
x
)
2
+b
(
x
x
)
(

)]
= aVar
[
x
]
= a
→
a =
.5*
2
=
0.707
; b =
1.22
Problem 2 ( 25 points)
1
Suppose that the time between failures of a structural component is modeled as an exponentially
distributed random variable.
You want to use the 10% quantile
x
10
[defined by
F
x
(
x
10
) = 0.10] as an
indication of how often the component should be tested.
You have the following 10 recorded
times between failures (in hrs):
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 Spring '05
 GeorgeKocur
 Environmental Engineering

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