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,

Thus, random variable X by itself has a normal distribution N(mX ,^2 X).
Similar calculations show that Y is also normal with distribution N(mY ,^2 Y ),
and is the correlation co efficient of X and Y. We thus see that
the five parameters contained in the bivariate density function fX Y (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 0 in Equation (7.27). We easily get

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 ,...,Xn. They are said to
be jointly normal if the associated jo int den sity function has the form

wheremT sthe
n n covariance matrix of X with [see Equations (4.34) and (4.35)]:

206 Fundamentals of Probability and Statistics for Engineers

fX…x†ˆ

Z 1

1

fXY…x;y†dyˆ

1

… 2 †^1 =^2 X

exp 

…xmX†^2

2 ^2 X

"#

; 1<x< 1 :… 7 : 28 †





ˆXY/XY

ˆ

fXY…x;y†ˆ

1

… 2 †^1 =^2 X

exp

…xmX†^2

2 ^2 X

)"

1

… 2 †^1 =^2 Y

exp 

…ymY†^2

2 ^2 Y

)"

ˆfX…x†fY…y†; … 7 : 29 †

fX 1 X 2 ...Xn…x 1 ;x 2 ;...;xn†ˆfX…x†

ˆ… 2 †n=^2 jj^1 =^2 exp

1

2

…xm†T^1 …xm†



;

1<x< 1 ; … 7 : 30 †

ˆ[m 1 m 2 ...mn]ˆ[EfX 1 gEfX 2 g...EfXng], andˆ[ij]i



ijˆEf…Ximi†…Xjmj†g: … 7 : 31 †