Page 128 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 111

Hence, as n !1,
2
2
XY t; s† e t ‡ts‡s † : 4:94†
Now, substituting Equation (4.94) into Equation (4.87) gives

1 Z 1 Z 1 j tx‡sy† t ‡ts‡s †
2
2
f XY x; y†ˆ e e dtds; 4:95†
4 2 1 1
which can be evaluated following a change of variables defined by

0
0
t ‡ s 0 t s 0
t ˆ p  ; s ˆ p  : 4:96†
2 2
The result is

2 2
1 x xy ‡ y
f XY x; y†ˆ p  exp : 4:97†
2 3 3
The above is an example of a bivariate normal distribution, to be discussed in
Section 7.2.3.
Incidentally, the joint 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

q XY t; s† 2 2
10 ˆ j ˆ j 2t s†e t ‡ts‡s † ˆ 0;
qt
t;sˆ0 t;sˆ0

01 ˆ j q XY t; s† ˆ 0:
qs
t;sˆ0
Similarly, the second moments are

2
2
20 ˆ EfX gˆ q XY t; s† ˆ 2;
qt 2
t;sˆ0
2
2
02 ˆ EfY gˆ q XY t; s† ˆ 2;
qs 2
t;sˆ0
2
11 ˆ EfXYgˆ q XY t; s† ˆ 1:
qt@s
t;sˆ0

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