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Problems 719
B–20 Let x be a random variable that has a Laplacian distribution. The Laplacian PDF is f(x) = (12b)
e -x-m > b , where b and m are real constants and b 0.
(a) Find the mean of x in terms of b and m.
(b) Find the variance of x in terms of b and m.
B–21 Referring to your solution for Prob. B–20, use MATLAB to plot the Laplacian PDF for m = 15
and s = 5.
B–22 Given the Gaussian PDF
1
>(2b
f(x) =
2 )
12pb e-(x-m)2
show that the variance of this distribution is b 2 .
★ B–23 In a manufacturing process for resistors, the values obtained for the resistors have a Gaussian
distribution where the desired value is the mean value. If we want 95% of the manufactured 1-kΩ
resistors to have a tolerance of ;10%, what is the required value for s?
B–24 Assume that x has a Gaussian distribution. Find the probability that
(a) |x - m| s.
(b) |x - m| 2s.
(c) |x - m| 3s.
Obtain numerical results by using MATLAB, or tables if necessary.
B–25 Show that
(a) Q(z) = 1 2 erfc a z 12 b.
(b)
(c)
B–26 Using MATLAB, plot Q(z) as defined by Eq. (B–60) and Prob. B–25.
B–27 For a Gaussian distribution, show that
(a)
Q(-z) = 1 - Q(z).
Q(z) = 1 2 c1 - erfa z
12 bd.
F(a) = 1
2 erfc a m - a
12s b.
(b) F(a) = 1 2
c1 + erfc a
a - m
12s bd.
B–28 Using MATLAB, plot the CDF for a Gaussian random variable where m = 10 and s = 2.
B–29 A noise voltage has a Gaussian distribution. The RMS value is 5 V, and the DC value is 1.0 V.
Find the probability of the voltage having values between -5 and +5 V.
★ B–30 Suppose that x is a Gaussian random variable with m = 5 and s = 0.6.
(a) Find the probability that x 1.
(b) Find the probability that x 6.
B–31 The Gaussian random variable x has a zero mean and a variance of 2. Let A be the event such that
|x| 3.
(a) Find an expression for the conditional PDF f(x|A).
(b) Plot f(x|A) over the range |x| 5.
(c) Plot f(x) over the range |x| 5 and compare these two plots.