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B–8 Functional Transformations of Random Variables 707
where B 0. For y 0, M = 1; and for y 0, M = 0. However, if y = 0, there are an infinite
number of roots for x (i.e., all x 0). Consequently, there will be a discrete point at y = 0 if the
area under f x (x) is nonzero for x 0 (i.e., for the values of x that are mapped into y = 0). Using
Eq. (B–68), we get
where
f x (y>B)
y 7 0
f y (y) = c B
s + P(y = 0) d(y)
0, y 6 0
0
P(y = 0) = P(x … 0) = f x (x) dx = F x (0)
L -q
(B–73)
(B–74)
Suppose that x has a Gaussian distribution with zero mean; then these equations reduce to
1 >(2B 2 s 2)
f y (y) = c 22pBs e-y2 , y 7 0
s + 1 2 d(y)
0, y 6 0
(B–75)
A sketch of this result is shown in Fig. B–10. For B = 1, note that the output is the same as
the input for x positive (i.e., y = x 0), so that the PDF of the output is the same as the PDF of
the input for y 0. For x 0, the values of x are mapped into the point y = 0, so that the PDF of
1
y contains a d function of weight at the point y = 0.
2
Proof. We will demonstrate that Eq. (B–68) is valid by partitioning the x-axis into
intervals over which h(x) is monotonically increasing, monotonically decreasing, or a
constant. As we have seen in Ex. B–10, when h(x) is a constant over some interval of x,
a discrete point at y equal to that constant is possible. In addition, discrete points in the
distribution of x will be mapped into discrete points in y, even in regions where h(x) is not
a constant.
Now we will demonstrate that the theorem, as described by Eq. (B–68), is correct
by taking the case, for example, where y = h(x) is monotonically decreasing for x x 0
and monotonically increasing for x x 0 . This is illustrated in Fig. B–11. The CDF for y
is then
where the + sign denotes the union operation. Then, using Eq. (B–4), we get
or
F y (y 0 ) = P(y … y 0 ) = P(x 1 … x … x 2 )
= P[(x = x 1 ) + (x 1 6 x … x 2 )]
F y (y 0 ) = P(x 1 ) + P(x 1 6 x … x 2 )
(B–76)
F y (y 0 ) = P(x 1 ) + F x (x 2 ) - F x (x 1 )
(B–77)