Page 227 - DMGT209_QUANTITATIVE_TECHNIQUES_II

Quantitative Techniques-II

Notes
Cov X,Y 
Consider equation (8), b  2

X
r   
Writing Cov(X, Y) = r ×   , we have b  X Y  r  Y
X Y 2
 X  X
The line of regression of Y on X, i.e Y = a + bX can also be written as
Ci i
or Y = Y bX  bX or Y  Y   b X  X  .... (11)
Ci i Ci i
 Y
or  Y  Y  = r   X  X  X  .... (12)
i
Ci
Line of Regression of X on Y

The general form of the line of regression of X on Y is X = c + dY , where X denotes the
Ci i Ci
predicted or calculated or estimated value of X for a given value of Y = Y and c and d are
i
constants. d is known as the regression coefficient of regression of X on Y.
In this case, we have to calculate the value of c and d so that
2
S = (X – X ) is minimised.
i Ci
As in the previous section, the normal equations for the estimation of c and d are
X = nc + dY .... (13)
i i
and X Y = cSY + dY 2 .... (14)
i i i i
Figure 11.2

Y
X c bY i
= +
ci
Y
i
Y X
ci i
c
O X

Dividing both sides of equation (13) by n, we have X  c dY.


This shows that the line of regression also passes through the point  X, Y . Since both the lines


of regression passes through the point  X,Y , therefore  X, Y is their point of intersection as
shown in Figure 11.3.
We can write c = X dY .... (15)
As before, the various expressions for d can be directly written, as given below.

 X Y  nXY
i
i
d = 2 2 .... (16)
 Y  nY
i

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