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By Jiruse M., Machek J.

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K. Let x1 À x2 and y1 À y2 be two sets of Cartesian coordinates, then the area of the x1 À x2 plane corresponding to a small rectangular d Sy ¼ d y1 Â d y2 is given by   @ ðiÞ k k X X @ y1 h1 ðy1 ; y2 Þ  d Sx ¼ jJk jd Sy ¼  ðiÞ i¼1 i¼1  @ h2 ðy1 ; y2 Þ  @ y1  @ ðiÞ  h1 ðy1 ; y2 Þ   @ y2 d Sy  @ ðiÞ  h2 ðy1 ; y2 Þ  @ y2 ð2:162Þ Dðx1 ; x2 Þ is a Jacobian of the transformation of each branch of the function Dðy1 ; y2 Þ x ¼ hðyÞ. Thus, it follows from eq. 161) that the transformation of the PDF is where J ¼   @ ðiÞ  h k X  @ y1 1 ðiÞ ðiÞ  pg ðy1 ; y2 Þ ¼ pn ðh1 ðy1 ; y2 Þ; h2 ðy1 ; y2 ÞÞ  @ ðiÞ i¼1 h  @ y1 2  @ ðiÞ  h  @ y2 1  @ ðiÞ  h  @ y2 2 ð2:163Þ In general, if a random n ¼ ½1 ; 2 ; .

Since rk l ¼ rl k , the covariation matrix R is a symmetric matrix and contains nðn þ 1Þ=2 independent elements. Altogether, nðn þ 1Þ=2 þ n parameters are needed to uniquely define Gaussian distribution. The conditional probability density can be easily obtained as pðx1 ; x2 ; . . ; xk jxkþ1 ; . . ; xn Þ ¼ pðX 1 jX2 Þ " # ðx À mX1 jX2 ÞT RÀ1 ðx À m Þ 1 X jX X1 jX2 1 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp À 2 k ð2Þ jRX1 jX2 j ð2:155Þ where R R ¼ 11 R21 ! R12 ¼ R11 À R12 RÀ1 R 22 R21 ; R22 X1 jX2 and mX1 jX2 ¼ m1 þ R12 RÀ1 22 ðX 2 À m2 Þ ð2:156Þ and the mean vector and the correlation matrix are split according to conditional and nonconditional variables.

169) can be further simplified if 1 and 2 are independent. In this case pn ðx1 ; x2 Þ ¼ p1 ðx1 Þp2 ðx2 Þ and ð1 p1 ðy Ç y2 Þp2 ðy2 Þd y2 ð2:170Þ p ðyÞ ¼ À1 In general, if there are n independent random variables 1 ; . . + n n n Y X Y ak k hexpðÀj u ak k Þi ¼ Âk ð j u ak Þ Â ð j uÞ ¼ hexpðÀj u Þi ¼ exp Àj u ¼ k¼1 k¼1 k¼1 ð2:172Þ Similarly, the expression for the log-characteristic function becomes É ð j uÞ ¼ ln  ð j uÞ ¼ n X ln Âk ð j u ak Þ ¼ k¼1 n X Ék ð j u ak Þ ð2:173Þ k¼1 The resulting distribution is called the composition of the distributions of each variable.

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