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720
Probability and Random Variables
Appendix B
B–32 Let x have a sinusoidal distribution with a PDF as given by Eq. (B–67). Show that the CDF is
0, a … -A
F(a) = e
1
p c p 2 + sin-1 a a A bd,
|a| … A
B–33 (a) If x has a sinusoidal distribution with the peak value of x being A, show that the RMS value is
s = A> 12. [Hint: Use Eq. (B–67).]
(b) If x = A cos c, where c is uniformly distributed between -p and +p, show that the RMS
value of x is s = A> 12.
★ B–34 Given that y = x 2 and x is a Gaussian random variable with mean value m and variance s 2 , find a
formula for the PDF of y in terms of m and s 2 .
B–35 x is a uniformly distributed random variable over the range -1 x 1 plus a discrete point at
x = 1 with PAx = 1 2 B = 1 2
4 .
(a) Find a mathematical expression for the PDF for x, and plot your result.
(b) Find the PDF for y, where
Sketch your result.
B–36 A saturating amplifier is modeled by
1, a Ú A
y = c
y = e x2 , x Ú 0
0, x 6 0
Ax 0 , x 7 x 0
Ax, |x| … x 0
-Ax 0 , x 6 -x 0
Assume that x is a Gaussian random variable with mean value m and variance s 2 . Find a formula
for the PDF of y in terms of A, x 0 , m, and s 2 .
B–37 Using MATLAB and your results for Prob. B–36, plot the PDF for the output of a saturating
amplifier with a Gaussian input if x 0 = 5, A = 10, m = 2, and s = 1.5.
★ B–38 A sinusoid with a peak value of 8 V is applied to the input of a quantizer. The quantizer characteristic
is shown in Fig. 3–8a. Calculate and plot the PDF for the output.
B–39 A voltage waveform that has a Gaussian distribution is applied to the input of a full-wave rectifier
circuit. The full-wave rectifier is described by y(t) = x(t), where x(t) is the input and y(t) is the
output. The input waveform has a DC value of 1 V and an RMS value of 2 V.
(a) Plot the PDF for the input waveform.
(b) Plot the PDF for the output waveform.
★ B–40 Refer to Example B–10 and Eq. (B–75), which describe the PDF for the output of an ideal diode
(half-wave rectifier) characteristic. Find the mean (DC) value of the output.
B–41 Given the joint density function,
f(x 1 , x 2 ) = e e-(1>2)(4x 1+x 2 ) , x 1 Ú 0, x 2 Ú 0
0, otherwise