Page 152 - DMTH404_STATISTICS

Statistics

Notes Note that E[{Y – E(Y)}] = 0, the sum of deviations of values from their arithmetic mean.

Remarks:
1. If X and Y are independent random variables, the right hand side of the above equation
will be zero. Thus, covariance between independent variables is always equal to zero.

2. COV(a + bX, c + dY) = bd COV(X, Y)
3. COV(X, X) = VAR(X)
II. Mean and Variance of a Linear Combination

)
+
Let Z f= ( ,X Y = aX bY be a linear combination of the two random variables X and Y, then
using the theorem of addition of expectation, we can write
+
=
 = E ( ) E (aX bY ) aE ( ) bE ( ) a  + b 
Z
+
Y
=
=
X
Z X Y
Further, the variance of Z is given by
2
2
2
)ù
 = E [Z E- (Z ] ) = E [aX bY+ - a  - b  ] = Eé ( a X - ) b+ (Y - Y û 2
Z X Y ë X
2
2
2
= a E (X - X ) + b 2 ( E Y - Y ) + 2abE ( X - X )( Y - Y )
2
2
2
2
= a  + b  + 2ab XY
X
Y
Remarks:
1. The above results indicate that any function of random variables is also a random variable.
2
2
2
2
2. If X and Y are independent, then  XY = 0 , \  = a  + b  2 Y
X
Z
2
2
2
2
2
2
2
2
2
2
3. If Z = aX - bY, then we can write  = a  + b  - 2ab XY . However,  = a  + b  , if
Z
Z
X
Y
Y
X
X and Y are independent.
4. The above results can be generalised. If X , X , ...... X are k independent random variables
1 2 k
2
with means  1 , 2 , ......  and variances  2 1 , 2 2 , ......  respectively, then
k
k
E (X  X  ....  X k ) =     ....   k
2
2
1
1
2
2
and Var (X  X  ....  X k ) =  +  + .... +  k 2
2
1
2
1
Notes
1. The general result on expectation of the sum or difference will hold even if the
random variables are not independent.
2. The above result can also be proved for continuous random variables.
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