Page 224 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 207

The superscripts T and 1 denote, respectively, matrix transpose and matrix
inverse. Again, we see that a joint normal distribution is completely specified
by the first-order and second-order joint moments.
It is instructive to derive the joint characteristic function associated with X.
As seen from Section 4.5.3, it is defined by
t 1 ; t 2 ; ... ; t n †ˆ t†
X 1 X 2 ...X n X
ˆ Efexp‰j t 1 X 1 ‡ ‡ t n X n †Šg
1 1 7:32†
Z Z
T
ˆ exp jt x†f x†dx;
X
1 1
which gives, on substituting Equation (7.30) into Equation (7.32),
1
T
T
t†ˆ exp jm t t t ; 7:33†
X
2
T
where t ˆ [t 1 t 2 t n ].
Joint moments of X can be obtained by differentiating joint characteristic
function X (t) with respect to t and setting t ˆ 0. The expectation
EfX X 2 m 2 X g, for example, is given by
m n
m 1
n
1
q m 1 ‡m 2 ‡ ‡m n

m 1
EfX X m 2 X gˆ j m 1 ‡m 2 ‡ ‡m n † t† 7:34†
m n
1 2 n m 1 m 2 m n X
qt qt qt n
1 2 tˆ0
It is clear that, since joint moments of the first-order and second-order
completely specify the joint normal distribution, these moments also determine
joint moments of orders higher than 2. We can show that, in the case when
random variables X 1 , X 2 , , X n have zero means, all odd-order moments of
these random variables vanish, and, for n even,
X
EfX 1 X 2 X n gˆ EfX m 1 X m 2 gEfX m 2 X m 3 g EfX m n 1 X m n g 7:35†
m 1 ;...;m n
The sum above is taken over all possible combinations of n/2 pairs of the
n random variables. The number of terms in the summation is (1)(3)(5)
(n 3)(n 1).

7.2.4 SUMS OF NORMAL RANDOM VARIABLES

We have seen through discussions and examples that sums of random variables
arise in a number of problem formulations. In the case of normal random
variables, we have the following important result (Theorem 7.4).

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