Fundamentals of Probability and Statistics for Engineers

Hence, as n

Now, substituting Equation (4.94) into Equation (4.87) gives

which can be evaluated following a change of variables defined by

The result is

The above is an example of a bivariate normal distribution, to be discussed in
Section 7.2.3.
Inciden tally, the jo int moments of X and Y can be readily found by means of
Equation (4.88). For large n, the means of X and Y, 10 and 01 ,are

Similarly, the second moments are

Expectations and Moments 111

!1,

XY…t;s†e…t

(^2) ‡ts‡s (^2) †
: … 4 : 94 †
fXY…x;y†ˆ

1

4 ^2

Z 1

1

Z 1

1

ej…tx‡sy†e…t

(^2) ‡ts‡s (^2) †
dtds; … 4 : 95 †
tˆ
t^0 ‡s^0

2
p ; sˆ
t^0 s^0

2
p : … 4 : 96 †
fXY…x;y†ˆ

1

2 



3

p exp

x^2 xy‡y^2

3



: … 4 : 97 †

10 ˆj

qXY…t;s†

qt

t;sˆ 0

ˆj… 2 ts†e…t

(^2) ‡ts‡s (^2) †
t;sˆ 0

ˆ 0 ;

01 ˆj

qXY…t;s†

qs

t;sˆ 0

ˆ 0 :

20 ˆEfX^2 gˆ

q^2 XY…t;s†

qt^2

t;sˆ 0

ˆ 2 ;

02 ˆEfY^2 gˆ

q^2 XY…t;s†

qs^2

t;sˆ 0

ˆ 2 ;

11 ˆEfXYgˆ

q^2 XY…t;s†

qt@s

t;sˆ 0

ˆ 1 :