Page 304 - DMTH404_STATISTICS

Statistics

Notes 2 2
 X  X   Y  Y 
i
i
2
2
 x'  and y' 
i
i
 2 X  2 Y
2   X  X  2 2   Y  Y  2
i
i
or  x'  2 and  y'  2
i
i
 
X Y
i 
2
2
From these summations we can write  x'  y'  n
i
1   X  X  Y  Y 
n i i 1  X  X  Y  Y  1
i
i
Also, r =         x' y' i
i
  n   n
X Y  X   Y 
Consider the sum x ' + y '. The square of this sum is always a non-negative number, i.e., (x '
i i i
2
+ y ') ³ 0.
i
Taking sum over all the observations and dividing by n, we get
1  x'  y'   0 or  x'  y'  2x' y'  0
1
2
2
2
n i i n i i i i
1 1 2
2
2
or  x'   y'   x' y'  0
i
i
i
i
n n n
or 1 + 1 + 2r  0 or 2 + 2r  0 or r  – 1 .... (11)
Further, consider the difference x ' - y '. The square of this difference is also non-negative,
i i
i.e., (x ' - y ') ³ 0.
2
i i
Taking sum over all the observations and dividing by n, we get
1 2 1 2 2
 x'  y'   or  x'  y'  2x' y'  0
0
n i i n i i i i
1 2 1 2 2
or  x'   y'   x' y'  0
i
i
i
i
n n n
or 1 + 1 - 2r  0 or 2 - 2r  0 or r  1 .... (12)
Combining the inequalities (11) and (12), we get - 1  r  1. Hence r lies between -1 and +1.
3. If X and Y are independent they are uncorrelated, but the converse is not true.
If X and Y are independent, it implies that they do not reveal any tendency of simultaneous
movement either in same or in opposite directions. In terms of figure 12.3, the dots of the
scatter diagram will be uniformly spread in all the four quadrants. Therefore,
  X  X  Y  Y  or Cov(X, Y) will be equal to zero and hence, r = 0. Thus, if X and Y are
i
i
XY
independent, they are uncorrelated.
The converse of this property implies that if r = 0, then X and Y may not necessarily be
XY
independent. To prove this, we consider the following data :
X 1 2 3 4 5 6 7
Y 9 4 1 0 1 4 9
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