*Notes to a video lecture on http://www.unizor.com*

__Linear Regression - Problem 1__

Consider a linear regression model described in the previous lecture:

**Y = a·X + b + ε**where independent variable

*is represented by sample data*

**X***x*,

_{1}*x*...

_{2}*x*

_{n}and observed values of dependent variable

*are*

**Y***y*,

_{1}*y*...

_{2}*y*.

_{n}We have introduced two averages of the sample data:

*Ave(x)=(x*

_{1}+x_{2}+...+x_{n})/n = U*Ave(y)=(y*

_{1}+y_{2}+...+y_{n})/n = VUsing new variables

*X*and

_{k}= x_{k}− U*Y*

_{k}= y_{k}− V(where index

*k*is from

*1*to

*n*)

we came up with the best possible value for a coefficient

*in the formula for linear regression as*

**a**

**a**= ΣX_{k}·Y_{k}/ ΣX_{k}²*Problem*

Using the sample averaging function

*Ave()*applied as

*Ave(x)=(x*

_{1}+x_{2}+...+x_{n})/n*Ave(y)=(y*

_{1}+y_{2}+...+y_{n})/n*Ave(xy)=(x*

_{1}y_{1}+...+x_{n}y_{n})/n*Ave(x²)=(x*

_{1}²+...+x_{n}²)/nprove that the expression for a coefficient

*in the formula for linear regression in terms of original sample data*

**a***x*and

_{k}*y*looks like

_{k}

**a**= [*Ave(xy)−Ave(x)Ave(y)] / [*

*Ave(x²)−Ave²(x)]*

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