Stat 135, Fall 2006
HOMEWORK 9 (due Friday 11/17)
You have two weeks to do these problems. If you decide to take a week off and just do
them in the second week, you will not find me helpful. On Tuesday 11/14 I will help you
with at most three problems, and on Thursday 11/16, I will help you with at most one part
of one problem.
1.
12.26.
2.
12.27.
3.
12.28.
4.
Let
X
and
Y
be two squareintegrable random variables.
That means
E
(
X
2
) and
E
(
Y
2
) are both finite. Define the
correlation
between
X
and
Y
to be
R
(
X, Y
)
=
E
X

E
(
X
)
SD
(
X
)
Y

E
(
Y
)
SD
(
Y
)
Let
X
*
be
X
in standard units, so that
X
*
= (
X

E
(
X
))
/SD
(
X
).
Let
Y
*
be
Y
in
standard units.
a)
Show that
R
(
X, Y
) =
R
(
Y, X
) =
R
(
X
*
, Y
*
). This means that the correlation between
two variables does not depend on the order of the variables nor on the units in which the
variables were measured.
b)
Show that

1
≤
R
(
X, Y
)
≤
1.
[Hint: The random variables (
X
*
+
Y
*
)
2
and (
X
*

Y
*
)
2
are nonnegative, therefore so
are their expectations.]
In what follows, you have two observations on each of
n
individuals. The observations
are (
x
1
, y
1
)
,
(
x
2
, y
2
)
, . . . ,
(
x
n
, y
n
). Define ¯
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 Spring '12
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 Regression Analysis, two weeks, regression line, regression estimate, Final Exam Score

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