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704
Probability and Random Variables
Appendix B
the oscillation builds up from a noise voltage that is present in the circuit.) In another oscillator
model, time might be considered to be a uniformly distributed random variable, where
c = v 0 t. Here, once again, we would have a sinusoidal distribution for x.
B–8 FUNCTIONAL TRANSFORMATIONS
OF RANDOM VARIABLES
As illustrated by the preceding sinusoidal distribution, we often need to evaluate the PDF for
a random variable that is a function of another random variable for which the distribution is
known. This is illustrated pictorially in Fig. B–8. Here the input random variable is denoted
by x, and the output random variable is denoted by y. Because several PDFs are involved,
subscripts (such as x in f x ) will be used to indicate with which random variable the PDF is
associated. The arguments of the PDFs may change, depending on what substitutions are
made, as equations are reduced.
THEOREM. If y = h(x), where h(·) is the output-to-input (transfer) characteristic of a
device without memory, † then the PDF of the output is
M
f x (x)
f y (y) = a `
(B–68)
i = 1 ƒ dy>dx ƒ x = x i = h 1 ī (y)
where f x (x) is the PDF of the input, x. M is the number of real roots of y = h(x). That is, the
inverse of y = h(x) gives x 1 , x 2 ,..., x M for a single value of y. · denotes the absolute value
and the single vertical line denotes the evaluation of the quantity at x = x i = h i ī (y).
Two examples will now be worked out to demonstrate the application of this theorem, and
then the proof will be given.
Example B–9 SINUSOIDAL DISTRIBUTION
Let
y = h(x) = A sin x
(B–69)
where x is uniformly distributed over -p to p, as given by Eq. (B–66). This is illustrated in
Fig. B–9. For a given value of y, say -A 6 y 0 6 A, there are two possible inverse values for x,
x
Input PDF,
f x (x), given
h(x)
Transfer characteristic
(no memory)
y=h(x)
Output PDF,
f x (y), to be evaluated
Figure B–8
Functional transformation of random variables.
† The output-to-input characteristic h(x) should not be confused with the impulse response of a linear
network, which was denoted by h(t).