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We have one independent variable X and a variable Y that is dependent
on
X.
Specifically, it is assumed that Y depends on X via the following
linear relationship: Y= b0
+ b1X + e
where b0 and
b1
are unknown constants and e
is an unknown random error component.
Thus given a value of X=Xi, Y
"tends" to take on a value Yi=b0
+ b1Xi - the actual
value varies around this by the random value e.
We assume the following about e
: ei ~ N(0,s2)
and cov(ei,ej)=0
for i¹ j.
Thus the error component follows a Normal distribution with a mean
of zero and a variance of s2 and
the errors associated with two separate observations i and j
are uncorrelated.
We take a set of n observations of X and Y, i.e., (X1,Y1), (X2,Y2),…, (Xn,Yn) and using this data set, we estimate the values of b0 by b0 and the value of b1by b1.
b1 =,
b0=
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