Page 151 - DMTH404_STATISTICS

Unit 10: Approximate Expressions for Expectations and Variance

Theorem 2. Notes

If X and Y are two independent random variables, then
E(X.Y) = E(X).E(Y)
Proof.

Let the random variable X takes values X , X , ...... X and the random variable Y takes values Y ,
1 2 m 1
Y , ...... Y such that P(X = X and Y = Y) = p (i = 1 to m, j = 1 to n).
2 n i j ij
m n
By definition (E XY = åå X Y p
)
j ij
i
i= 1 j= 1
Since X and Y are independent, we have p = P . ¢
P
j
ij
i
m n m n
)
\ (E XY = å å X Y . P P =  å X P ´ YP
i å
i j i j i j j
i= 1 j= 1 i= 1 j= 1
= E(X).E(Y).
The above result can be generalised. If there are k independent random variables X , X , ...... X ,
1 2 k
then
E(X . X . ...... X ) = E(X ).E(X ). ...... E(X )
1 2 k 1 2 k
10.2.4 Expectation of a Function of Random Variables
a
Let f X,Yf be a function of two random variables X and Y. Then we can write
m n
E f ( [ X ,Y )] = åå f (X i ,Y j ) p ij
i= 1 j= 1
I. Expression for Covariance

E
=
Y -
As a particular case, assume that (Xf i ,Y j ) (X= i -  X )( j  Y ) , where ( )E X =  X and ( )  Y
Y
m n
)ù = åå
Eé
Thus, (X -  X )(Y -  Y û (X -  X )(Y -  Y ) p ij
ë
i
j
i= 1 j= 1
The above expression, which is the mean of the product of deviations of values from their
respective means, is known as the Covariance of X and Y denoted as Cov(X, Y) or  XY . Thus, we
can write
Cov ( ,X Y = Eé ë (X -  X )(Y -  Y û
)
)ù
An alternative expression of Cov(X, Y)
X
Cov ( , ) = Eé ë {X E- ( ) }{Y E- (Y } ) ù û
X
Y
= é ë { . Y E- (Y } ) - ( E X { ). Y E- (Y } ) ù û
E X
=E X.Y - X.E(Y) =E(X.Y)- E(X).E(Y)

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