Page 106 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 89

Proof of Property 4.2: to show Property 4.2, let t and u be any real quantities

and form
2
t; u†ˆ Ef‰t X m x †‡ u Y m Y †Š g
2
2
ˆ 20 t ‡ 2 11 tu ‡ 02 u :
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
20 02 2 0; 4:25†
11
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 , and b 2
with a 1 > 0 and a 2 > 0, we can easily verify that
a 1 X ‡ b 1 ; a 2 Y ‡ b 2 †ˆ X; Y†: 4:26†

Property 4.3. If X and Y are independent, then

11 ˆ 0 and ˆ 0: 4:27†

Proof of Property 4.3: let X and Y be continuous; their joint moment 11 is
found from
Z 1 Z 1
11 ˆ EfXYgˆ xyf x; y†dxdy:
XY
1 1
If X and Y are independent, we see from Equation (3.45) that
f x; y†ˆ f x†f y†;
XY X Y
and
1 1 1 1
Z Z Z Z
11 ˆ xyf x†f y†dxdy ˆ xf x†dx yf y†dy
X Y X Y
1 1 1 1
ˆ m X m Y :
Equations (4.23) and (4.24) then show that 11 ˆ 0 and ˆ 0. A similar result
can be obtained for two independent discrete random variables.

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