Proof of relationship between xi, yi and x, y in linear regression

The given proof shows the relationship between xi, yi and x, y in linear regression. It also explains the concept of simple linear regression model.

2. Show that the following relationships holds
∑
i=i
n
( xi −x ) ( yi − y )=∑
i=1
n
xi ( yi− y ) …………………………. (xx)
Proof
Expand the right hand side of the equation (xx) as follows
∑
i=1
n
( xi −x ) ( yi − y )=∑
i=1
n
xi ( yi− y ) −∑
i=1
n
x ( yi− y ) ……………… (i)
Consider the last term
∑
i=1
n
x ( yi− y ) =x ∑
i =1
n
( yi− y )
∑
i=1
n
x ( yi− y ) =x {∑
i=1
n
yi−∑
i=1
n
y }
∑
i=1
n
x ( yi− y ) =x {∑
i=1
n
yi−n y } …………………………………………. (ii)
But,
y=
∑
i=1
n
yi
n
, implying n y=∑
i=1
n
yi…………………………………………...... (iii)
Substitute (iii) in equation (ii) to obtain
∑
i=1
n
x ( yi− y ) =x {∑
i=1
n
yi−∑
i=1
n
yi } = 0
Now equation (i) becomes
∑
i=1
n
( xi −x ) ( yi − y )=∑
i=1
n
xi ( yi− y ) . Hence proved
3. Show that the following relationships holds
∑
i=1
n
yi =∑
i=1
n
^yi
Proof

We know that ei = yi− ^yi
Taking summation on both sides
∑
i=1
n
ei=∑
i=1
n
( yi− ^yi )
∑
i=1
n
ei=∑
i=1
n
yi−∑
i=1
n
^yi ……………………………………………… (i)
But, ∑
i=1
n
ei=¿ 0
Equation (i) becomes
0 ¿ ∑
i=1
n
yi−∑
i=1
n
^yi
Then, ∑
i=1
n
yi =∑
i=1
n
^yi. Hence proved.
4. Show that for the simple linear regression model, the point ( x , y) is always on the
regression line.
Proof
For simple linear regression yi= ^yi + ^ei and ^yi= ^b0 + ^b1 xi
But, y=
∑
i=1
n
yi
n =
∑
i=1
n
( ^yi + ^ei )
n
y=
∑
i=1
n
( ^yi + ^ei )
n
y=
∑
i=1
n
^yi
n −
∑
i=1
n
^ei
n
. ……………………………………….. (i)
Since, ∑
i=1
n
^ei=0 equation (i) becomes

y=
∑
i=1
n
^yi
n
y=
∑
i=1
n
( ^b0 + ^b1 xi )
n
y=
∑
i=1
n
^b0
n +
∑
i=1
n
^b1 xi
n
y= ^b0 + ^b1
∑
i=1
n
xi
n
……………………………………… (ii)
Since, x=
∑
i=1
n
xi
n
equation (ii) becomes
y= ^b0 + ^b1 x. Therefore, the regression line goes through point ( x , y).