Stat 312: Lecture 19
Linear Regression
Moo K. Chung
[email protected]
Nov 30, 2004
Concepts
1. Let
x
be the speed of a car and
y
be the distance the car
traveled in an hour hour. Then we have model
y
=
β
0
+
β
1
x.
Suppose we have
n
paired measurements
(
x
i
,y
i
)
,i
=
1
,
···
,n
. Since all measurement are supposed to be
noisy, we introduce a noise term
²
in the above equation.
Our modiﬁed stochastic model is
y
=
β
0
+
β
1
x
+
²,
where
²
∼
N
(0
,σ
2
)
. Since
²
is a random variable, we
use
Y
instead of
y
for convenience:
Y
=
β
0
+
β
1
x
+
².
Note that
E
Y
=
β
0
+
β
1
x
and
V
Y
=
σ
2
.
2. Equivalently we can write the above linear model for
each paired measurement
(
x
i
,y
j
)
:
Y
j
=
β
0
+
β
1
x
j
+
²
j
,
where
y
j
is the observed value of random variable
Y
j
and
²
j
∼
²
. Note that
E
Y
j
=
β
0
+
β
1
x
j
. Let
ˆ
β
0
,
ˆ
β
1
be
estimators of
β
0
,β
1
. Then the
predicted values
or ﬁtted
values are given by
ˆ
y
j
=
ˆ
β
0
+
ˆ
β
1
x
j
.
The differences between the observations
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This note was uploaded on 01/31/2008 for the course STAT 312 taught by Professor Chung during the Fall '04 term at University of Wisconsin.
 Fall '04
 Chung
 Statistics, Linear Regression

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