Stat 312: Lecture 17
Simple Linear Model
Moo K. Chung
[email protected]
March 27, 2003
Concepts
1. Deterministic model:
y
=
β
0
+
β
1
x.
For example,
x
is the speed of a car and
y
is the distance
the car traveled in 1 hour. This is an unrealistic model
because the car can not maintain the absolute constant
speed. So we introduce a noise term
±
in the above equa
tion.
2. Stochastic model: let
±
be a random noise with
±
= 0
and
Var
±
=
σ
2
. Instead of the deterministic model we
will have
y
=
β
0
+
β
1
x
+
±
. Since
±
is a random variable,
we use
Y
instead of
y
:
Y
=
β
0
+
β
1
x
+
±.
Note that
Y
=
β
0
+
β
1
x
and
Var
Y
=
σ
2
.
3. Given paired data
(
x
1
, y
1
)
,
···
,
(
x
n
, y
n
)
, we have a lin
ear stochastic relationship
Y
j
=
β
0
+
β
1
x
j
+
±
j
,
where
y
j
is the observed value of a random variable
Y
j
and
±
j
∼
±
. Note that
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
ˆ
y
j
=
ˆ
β
0
+
ˆ
β
1
x
j
are estimators of
Y
j
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This note was uploaded on 01/31/2008 for the course STAT 312 taught by Professor Chung during the Spring '04 term at University of Wisconsin.
 Spring '04
 Chung
 Statistics

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