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708
Probability and Random Variables
Appendix B
y = h(x)
y o
x 1 x 0
x 2
x
Figure B–11 Example of a function h(x) that monotonically decreases for x x 0
and monotonically increases for x x 0 .
The PDF of y is obtained by taking the derivative of both sides of this equation:
dF y (y 0 )
dy 0
= dF x(x 2 ) dx 2
dx 2 dy 0
- dF x(x 1 )
dx 1
dx 1
dy 0
(B–78)
where dP(x 1 )dy 0 = 0 since P(x 1 ) is a constant. Because
Eq. (B–78) becomes
dF x (x 2 )>dx 2 = f x (x 2 ) and dF x (x 1 )>dx 1 = f x (x 1 )
f y (y 0 ) = f x(x 2 )
dy 0 >dx 2
+
f x (x 1 )
-dy 0 >dx 1
(B–79)
At the point x = x 1 , the slope of y is negative, because the function is monotonically
decreasing for x x 0 ; thus, dy 0 dx 1 0, and Eq. (B–79) becomes
f y (y 0 ) =
M = 2
a
i = 1
f x (x)
ƒ dy 0 >dxƒ
`
x = x i = h i -1 (y 0 )
(B–80)
When there are more than two intervals during which h(x) is monotonically increasing or
decreasing, this procedure may be extended so that Eq. (B–68) is obtained.
In concluding this discussion on the functional transformation of a random variable, it
should be emphasized that the description of the mapping function y = h(x) assumes that the
output y, at any instant, depends on the value of the input x only at that same instant and not
on previous (or future) values of x. Thus, this technique is applicable to devices that contain
no memory (i.e., no inductance or capacitance) elements; however, the device may be nonlinear,
as we have seen in the preceding examples.