Page 326 - DMTH404_STATISTICS

Statistics

Notes Dividing numerator and denominator of equation (6) by n we have

1
å ( X - X Y - Y ) )
i
)( i
n Cov (X Y,
b = = .... (8)
1 2 s 2
å ( X - X ) X
i
n
The expression for b, which is convenient for use in computational work, can be written from
equation (5) is given below:
)( Y
å
å X i å Y i ( X i å i )
å X Y - n × å X Y -
i i
i i
b = n n = n 2
2
å
2 æ å X i ö 2 ( X i )
å X - n ç ÷ å X -
i
è n ø i n
Multiplying numerator and denominator by n, we have
)( Y
( X
n X Y - å i å i )
å
i i
b = 2 2 .... (9)
( X
å
n X - å i )
i
To write the shortcut formula for b, we shall show that it is independent of change of origin but
not of change of scale.
As in case of coefficient of correlation we define
X - A Y - B
i
i
u = and v =
i
i
h k
or X = A + hu and Y = B + kv
i i i i
 X = A +hu and Y = B +kv
b
also d X - Xi = h u - ug and Y - Y = k v - vg
b
i
i
i
i
Substituting these values in equation (6), we have
u -
u -
hkå ( i u v - ) v kå ( i u v - ) v
)( i
)( i
b = =
2 2 2
u -
u -
h å ( i ) u hå ( i ) u
é ù
)( v
å
( u
k n u v - å i å i )
i i
= ê 2 ú .... (10)
2
h ê n u - å ) ú
( u
å
ë i i û
(Note: if h = k they will cancel each other)
a
Cov X,Yf
Consider equation (8), b =
s X 2
r.s s s
Writing Cov (X Y, ) = r.s s , we have b = X Y = r × Y
X
Y
s 2 s
X X
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