Page 223 - Fundamentals of Probability and Statistics for Engineers

206 Fundamentals of Probability and Statistics for Engineers

Let us determine the marginal density function of random variable X. It is
given by, following straightforward calculations,

" #
Z 1 1 x m X † 2
f x†ˆ f x; y†dy ˆ exp ; 1 < x < 1: 7:28†
X XY 1=2 2
1 2 † X 2 X
Thus, random variable X by itself has a normal distribution N(m X , 2 X ).
Similar calculations show that Y is also normal with distribution N(m Y , 2 Y ),
is the correlation coefficient of X and Y . We thus see that
and ˆ XY / X Y
the five parameters contained in the bivariate density function f XY (x, y) repre-
sent five important moments associated with the random variables. This also
leads us to observe that the bivariate normal distribution is completely char-
acterized by the first-order and second-order joint moments of X and Y .
Another interesting and important property associated with jointly normally
distributed random variables is noted in Theorem 7.3.
Theorem 7.3: Zero correlation implies independence when the random vari-
ables are jointly normal.
Proof of Theorem 7.3: let ˆ 0 in Equation (7.27). We easily get

" 2 #) " 2 #)
1 x m X † 1 y m Y †
f XY x; y†ˆ 1=2 exp 2 1=2 exp 2
2 † X 2 X 2 † Y 2 Y
ˆ f x†f y†; 7:29†
X Y
which is the desired result. It should be stressed again, as in Section 4.3.1, that
this property is not shared by random variables in general.
We have the multivariate normal distribution when the case of two random
variables is extended to that involving n random variables. For compactness,
vector–matrix notation is used in the following.
Consider a sequence of n random variables, X 1 , X 2 ,..., X n . They are said to
be jointly normal if the associated joint density function has the form

f x 1 ; x 2 ; ... ; x n †ˆ f x†
X 1 X 2 ...X n X
1
T
1
ˆ 2 † n=2 j j 1=2 exp x m† x m† ;
2
1 < x < 1; 7:30†
T
where m ˆ [m 1 m 2 ... m n ] ˆ [EfX 1 g EfX 2 g ... EfX n g], and ˆ [ ij ] i s the
n n covariance matrix of X with [see Equations (4.34) and (4.35)]:

ij ˆ Ef X i m i † X j m j †g: 7:31†

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