Fundamentals of Probability and Statistics for Engineers

206 Fundamentals of Probability and Statistics for EngineersLet us determine the marginal density function of random variable X. It isgiven by, following straightforward calculations,Z " #11 …x m X † 2f X …x† ˆ f XY …x; y†dy ˆ exp; 1 < x < 1: …7:28†1…2† 1=2 X2Thus, random variable X by itself has a normal distribution N(m X , X ).2Similar calculations show that Y is also normal with distribution N(m Y , Y ),and ˆ XY / X Y is the correlation coefficient of X and Y . We thus see thatthe five parameters contained in the bivariate density function f XY (x,y) representfive important moments associated with the random variables. This alsoleads us to observe that the bivariate normal distribution is completely characterizedby the first-order and second-order joint moments of X and Y .Another interesting and important property associated with jointly normallydistributed random variables is noted in Theorem 7.3.Theorem 7.3: Zero correlation implies independence when the random variablesare jointly normal.Proof of Theorem 7.3: let ˆ 0 in Equation (7.27). We easily getf XY …x; y† ˆ2 2 Xwhich is the desired result. It should be stressed again, as in Section 4.3.1, thatthis property is not shared by random variables in general.We have the multivariate normal distribution when the case of two randomvariables 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 tobe jointly normal if the associated joint density function has the formwhere m T ˆ [m 1 m 2 ... m n ] ˆ [EfX 1 g EfX 2 g ... EfX n g], and ˆ [ ij ] isthen n covariance matrix of X with [see Equations (4.34) and (4.35)]:2 2 X " #) " #)1 …x m X † 2 1 …y m Y † 2expexp…2† 1=2 X …2† 1=2 Yˆ f X …x†f Y …y†;f X1 X 2 ...X n…x 1 ; x 2 ; ...; x n †ˆf X …x†ˆ…2† n=2 jj 1=2 exp2 2 Y12 …x m†T 1 …x m† ;…7:29†1 < x < 1; …7:30† ij ˆ Ef…X i m i †…X j m j †g: …7:31†TLFeBOOK