Unformatted text preview: 218 Chapter 4 One Random Variable Problems 219 \ 4.21. (.1) Find and plot the pdf in Problem 4.8.
(b) Use the pdt to find the probabilities of the events: { X > a} and {X > 2a}:
4.22. (a) Find and plot the pdf in Problem 4.12.
(b) Use the pdf to find PI~1 s X < 0.25].
4.23. (a) Find and plot the pdf in Problem 4.13.
(b) Use the pdl to find PIX : O], PCX > 8].
4.24. (a) Find and plot the pdf of the random variable in Problem 4.14.
(b) Use the pdf to calculate the probabilities in Problem 4.14b.
4.25. Find and plot the pdf of the Weibull random variable in Problem 4.15 a.
4.26. Find the cdf of the Cauchy random variable which has pdf: 57' (a) 1n Problem 4.7,find FZ(zIb/4 E Z S [9/2) and fZ(z I b/4 E Z S [9/2)
(in Find FZ(zIB) and fz(zI a). WhereB = {x > b/Z}.
'38, A binary transmission system sends a “0” bit using a *1 voltage signal and a “1” bit by transmitting a +1 . The received signal is corrupted by noise N that has a Laplacian distri—
bution With parameter a. Assume that “0” bits and “1” bits are equiprobablg (a) Find the pdf'of the received signal Y = X + N, where X is the transmitted signal,
given that a “0" was transmitted; that a “1" was transmitted. (b) Suppose that the receiver decides a “0” was sent if Y < 0. and a “l ” was sent if Y 2 0. What is the probability that the receiver makes an error given that a +1 was
transmitted? a —l was transmitted? (c) What is the overall probability of error? a/n‘
V ——00 < x < oo,
’} 1
x‘ + a!“ fX(x) = cction 4.3: The Expected Value of X 39. Find the mean and variance of X in Problem 4.17: Find the mean and variance of X in Problem 4.18. Find the mean and variance of Y. the distance from the dart to the origin. in Problem 4.19.
. Find the mean and variance of Z . the minimum of the coordinates in a square. in Problem 4.20.
Find the mean and variance of X = (1 — {)‘1/2 in Problem 4.21. Find E[X] using Eq. (4.28).
Find the mean and variance of X in Problems 4.12 and 4.22. Find the mean and variance of X in Problems 4.13 and 4.23. Find E[X] using Eq. (4.28). Find the mean and variance of the Gaussian random Variable b dire ct . . .
‘ t ’ inte rat
Eqs. (4.27) and (4.34), 3 g ion or Prove Eqs. (4.28) and (4.29).
Find the variance of the exponential random variable. 4.27. A voltage Xis uniformly distributed in the set {#3. *2. . . . , 3, 4}.
(3) Find the pdf and cdf of the random variable X.
(b) Find the pdl and cdf of the random variable Y = f2X2 + 3.
(c) Find the pdf and cdf of the random variable W = cos(77X/8).
((1) Find the pdf and cdf of the random variable Z = cos2(er/8).
4.28. Find the pdf and edl' of the Zipf random variable in Problem 3.70. 4.29. Let C be an event for which PC] > 0. Show that F X(xI( ) satisfies the eight prope
a cdf. N 4.30. (a) In Problem 4.13, find FX(x C) where C = {X > 0}.
(b) Find FX(x C) where C = {X : 0}. ,
4.31. (a) In Problem 4.10, find ["X(x I B) where B 2 {hand does not stop at 3, 6.9
o’clock}. 
(b) Find FX(x I B”).
4.32. In Problem 4.13,1'ind fX(xI B) and FX(x I B) Where B 2 {X > 0.25}.
4.33. Let X be the exponential random variable. ,
(3) Find and plot FX(x I X > I). How does FX(x I X > t) differ from ["X(x)T
(b) Find and ploth/(X I X > t).
(c) Show that P[X > t + x I X > I] = PIX > x]. Explain why this is called
oryless property.
4.34. The Pareto random variable X has cdf: . (a) Show that the mean of the Weibull random variable in Problem 4.15 is P(1 + 1/ ,8)
where P(x) is the gamma function defined in Eq. (4.56). (b) Find the second moment and the variance of the Weibull random variable. Explain Why the mean of the Cauchy random variable does not exist. Show that EIX] does not exist for the Pareto random variable with a = 1 and xm : 1.
Verify Eqs. (4.36). (4.37). and (4.38). Let Y : A cos(cur) + ‘c Where A has mean m and variance (72 and w and c are constants.
Find the mean and variance of Y. Compare the results to those obtained in Example 4.15.
A limiter is shown in Fig. P4.2. gtx) (a) Find and plot the pdf of X.
(b) Repeat Problem 4.33 parts a and b for the Pareto random variable.
(c) What happens to PIX > t + x I X > t] astbecomes large? Interpret this
4.35. (3) Find and plot FX(x a s X s b). Compare FX(xIa s x s b) to FX(x).
(b) Find and plot fX(x u S X g b).
4.36. In Problem 4.6.find FR(r I R > 1) and fR(r I R > 1). ' (4' FIGURE P42 ...
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