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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Civil and Environmental Engineering
1.017 Computing and Data Analysis for Environmental Applications
Quiz 2
Thursday, November 7, 2002
Please answer all questions on a separate piece(s) of paper with your name clearly identified:
Problem 1 ( 60 points – you will receive partial credit for each part successfully completed)
Consider a random variable
x
with the triangular PDF shown below:
f
x
(
x
)
(0,2/
a
)
(0, 0)
(
a
, 0)
x
Note that this distribution has one unknown parameter
a
. Its mean and variance are:
]
[
E
x
=
3
a
]
[
Var
x
=
18
2
a
You will receive 15 extra credit points if you prove that these expressions are correct
.
Extra Credit
:
E[x]=
∫
a
0
xf(x)dx=
∫
[(2/a
2
)x
2
+2x/a]dx = a/3
Var[x] =
∫
a
0
(xmu)
2
f(x)dx =
∫
(xa/3)
2
(2/a2x/a
2
)dx = a
2
/18
(work in between is required)
Suppose that you obtain a random sample of 8
x
values: [
x
1
,
x
2
, …,
x
8
].
Perform the following steps:
1
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a)
Devise an unbiased estimator
a
ˆ for the parameter
a
that depends only on the sample
1
8
mean
m
x
=
∑
x
i
.
i
=
1
b)
Derive the mean and variance of this estimator.
Is the estimator consistent?
Why?
c)
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This note was uploaded on 11/29/2011 for the course CIVIL 1.00 taught by Professor Georgekocur during the Spring '05 term at MIT.
 Spring '05
 GeorgeKocur
 Environmental Engineering

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