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Department of Civil and Environmental Engineering
1.017 Computing and Data Analysis for Environmental Applications
Practice Quiz 3
December 5, 2001
Please answer all questions on a separate piece(s) of paper with your name clearly identified:
Problem 1 (15 points)
Answer each of the following questions in a few sentences:
a) Suppose that
y
is a random variable (e.g nitrate concentration in a lake).
Also, suppose that
you have taken 20 nitrate measurements
y
1
,
y
2
, …,
y
20
at various times and locations.
Explain the difference between
E
[
y
], the expectation of
y
, and
m
y
, the sample mean of
y
.
b) Define the concept of a random sample and explain why it is useful
c) Why is the sample mean of
y
1
,
y
2
, …,
y
20
a “good’ estimate of
E
[
y
]?
Solution:
a)
The random variable
y
is characterized by its probability density, which describes the
likelihood of obtaining an observation in a given range (or interval) of values.
The
expectation
E
[
y
] is the first moment of this density.
It is a property of the density rather
than any particular set of data.
The sample mean
m
y
is the arithmetic average of a
particular set of data and is derived from the data rather than the density.
b) A sample is a set of measurements
y
1
,
y
2
, …,
y
n
of some random variable
y
with a
probability density
f
y
(
y
).
The
n
measured values can be viewed as particular outcomes
associated with
n
related random variables, one for each member of the sample.
Generally
speaking, these
n
random variables are described by a multivariate probability distribution.
If the sample is random, the
y
i
’s are independent and all have the same marginal probability
density
f
y
(
y
) as the original variable
y
.
These properties greatly simplifies the task of
deriving the probability distributions of estimates computed from the sample.
c) The sample mean is a good estimate of
E
[
y
] because it is unbiased [
E
(
m
y
)=
E
(
y
)] and
consistent [
Var
(
m
y
) goes to zero as the sample size approaches infinity]. Together, these
properties imply that the sample mean converges to the expected value of
y
as the sample
size increases.
1
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 Spring '05
 GeorgeKocur
 Environmental Engineering

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