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B–9 Multivariate Statistics 713
x 1
y 1
x
x 2
x 3
•••
y = h(x)
Transfer characteristic
(no memory)
y 2
y 3
•••
y
x N
y N
Figure B–13
Multivariate functional transformation of random variables.
THEOREM. Let y = h(x) denote the transfer characteristic of a device (no memory)
that has N inputs, denoted by x = (x 1 , x 2 ,..., x N ); N outputs, denoted by y = (y 1 , y 2 ,..., y N );
and y i = h i (x). That is,
y 1 = h 1 (x 1 , x 2 , Á , x N )
y 2 = h 2 (x 1 , x 2 , Á , x N )
o
y N = h N (x 1 , x 2 , Á , x N )
(B–98)
Furthermore, let x i , i = 1, 2,..., M denote the real roots (vectors) of the equation y = h(x). The
PDF of the output is then
f y
(y) = a
M
i=1
f x (x)
|J(y>x)|
`
x = x i = h i -1 (y)
(B–99)
where |·| denotes the absolute value operation and J(y/x) is the Jacobian of the coordinate
transformation to y from x. The Jacobian is defined as
0h 1 (x)
0x 1
0h 2 (x)
J¢ y x ≤ = Det H 0x 1
o
0h N (x)
0x 1
0h 1 (x)
0x 2 p
0h 1 (x)
0x N
0h 2 (x) 0h 2 (x)
0x 2 p 0x N X
o
o
0h N (x)
p 0h N(x)
0x 2 0x N
(B–100)
where Det[·] denotes the determinant of the matrix [·].
A proof of this theorem will not be given, but it should be clear that it is a generalization
of the theorem for the one-dimensional case that was studied in Sec. B–8. The coordinate transformation
relates differentials in one coordinate system to those in another [Thomas, 1969]:
dy 1 dy 2
Á dy N = J¢ y x ≤ dx 1 dx 2
Á dx N
(B–101)