Page 342 - DMTH404_STATISTICS

Statistics

Notes 23.5 Mean and Variance of 'e ' values
i

(i) M ean of e values
i
We know that e = Y - Y .
i i Ci
Taking sum over all the observations, we have

i å
0
Y -
å e = å ( i Y Ci ) = å Y - Y = [from equation (1)]
i
Ci
 Mean of e values is equal to zero.
i
(ii) Variance of e values
i
The variance of e values, in case of regression of Y on X, is given by
i
1 2 1 2
2
S Y X = å ( i ) 0 = å ( i Y Ci ) .... (2)
e -
Y -
.
n n
2
å Y - Y ) is the magnitude of unexplained variation in Y]
[Note that ( i Ci
1 2
2
S Y X = å é ( i Y ) ( b X- i - X ) ù
Y -
.
n ë û
2
Y -
å ( i Y ) 2 b å ( X - X ) 2 b 2 å ( X - X Y - Y )
)( i
i
i
= + -
n n n
2 2 2 2 2 2 2
×
= s + b s - 2 b bs X = s - b s X
X
Y
Y
2 2 2 2 2
= s - r s = s Y ( 1- r )
Y
Y
Similarly, it can be shown that the mean of e' (= X - X ) values, in case of regression of X on Y,
i i Ci
is also equal to zero. Further, their variance, i.e.,
2
S X Y = s 2 X ( 1- r 2 )
.
Alternatively equation (2) can be written as
1 1
Y
Y
S 2 Y X = å Y - Y Y = ë éå i 2 - aå i - bå X Y ù û
.
i i
) i
ci
( i
n n
Similarly, we can write
1
S 2 = éå X 2 - cå X - då X Y ù
.
X Y ë i i i i û
n
Remarks:
The above expressions for the variance are based on the following:
å(Y – Y ) = å(Y – Y )(Y – Y )
2
i ci i ci i ci
= å(Y – Y )Y – å(Y – Y )Y
i ci i i ci ci
It can be shown that the last term is zero.
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