Fundamentals of Probability and Statistics for Engineers

P roof of P ropert y 4. 2: to show Property 4.2, let t and u be any real quantities
and form

Since the expectation of a nonnegative function of X and Y must be non-
negative, (t,u) is a nonnegative quadratic form in t and u, and we must
have

which gives the desired result.
The normalization of the covariance through Equation (4.24) renders a
useful substitute for 11. Furthermore, the correlation coefficient is dimension-
less and independent of the origin, that is, for any constants a 1 ,a 2 ,b 1 ,andb 2
with a 1 >0anda 2 > 0, we can easily verify that

P ropert y 4. 3. If X and Y are independent, then

P roof of P ropert y 4. 3: let X and Y be continuous; their jo int moment 11 is
found from

If X and Y are independent, we see from Equation (3.45) that

and

Equations (4.23) and (4.24) then show that similar result
can be obtained for two independent discrete random variables.

Expectations and Moments 89

…t;u†ˆEf‰t…Xmx†‡u…YmY†Š^2 g

ˆ 20 t^2 ‡ 2  11 tu‡ 02 u^2 :

 20  02 ^211  0 ;… 4 : 25 †





…a 1 X‡b 1 ;a 2 Y‡b 2 †ˆ…X;Y†: … 4 : 26 †

 11 ˆ0 and ˆ 0 : … 4 : 27 †

11 ˆEfXYgˆ

Z 1

1

Z 1

1

xyfXY…x;y†dxdy:

fXY…x;y†ˆfX…x†fY…y†;

11 ˆ

Z 1

1

Z 1

1

xyfX…x†fY…y†dxdyˆ

Z 1

1

xfX…x†dx

Z 1

1

yfY…y†dy

ˆmXmY:

 11 ˆ0 andˆ0. A