Unformatted text preview: 218 Chapter 4 One Random Variable Problems 219 I in? Find and plot the pdf in Problem 4.8.
Use the pdf to find the probabilities of the events: {X > a} and {X > 2a}
Find and plot the pdf in Problem 4.12.
Use the pdf to find P[i1 S X < 0.25].
4.23. (a) Find and plot the pdf in Problem 4.13. (b) Use the pdf to find I’EX : 0]. P[X > 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 4.15a.
4.26. Find the cdf of the Cauchy random variable which has pdf: , .37_ (a) in Problem 4.7, find Fz(z I b/4 E Z S b/2) and fZ(z I [9/4 S Z 5 17/2).
(1,) Find Fz(le) and fz(zB). where B : {x > b/2}.
4.33 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 (1. Assume that “0” bits and “1 ” bits are equiprobable. (3) Find the pdf of the received signal Y = X + N, Where X is the transmitted signal,
given that a “0” was tra nsmitted; that a “1 ” was transmitted. (b) Suppose that the receiver decides a “0” was sent if Y < 0, and a “1" was sent if
Y 2 0. What is the probability that the receiver makes an error given that a +1 was
transmitted? a —1 was transmitted? (c) What is the overall probability of. error? [Y/7T
fax) 2 ﬂ —oc < x < 00.
x‘ + a‘ Section 4.3: The Expected Value of X 4.39. Find the mean and variance of X in Problem 4.17. 4.40. Find the mean and variance of X in Problem 4.18. 4.41. Find the mean and variance of Y. the distance from the dart to the origin, in Problem 4.19.
4.42. Find the mean and variance of Z, the minimum of the coordinates in a square, in Problem 4.20.
4.43. Find the mean and variance of X = (1 — J) 1/2 in Problem 4.21. Find E [X] using Eq. (4.28).
4.44. Find the mean and variance of X in Problems 4.12 and 4.22. 4.45. Find the mean and variance of X in Problems 4.13 and 4.23. Find E[X] using Eq. (4.28).  4.46. Find the mean and variance of the Gaussian random variable by direct integration of
Eqs. (4.27) and (4.34). 4.47. Prove Eqs. (4.28) and (4.29). . Find the variance of the exponential random variable. . (a) Show that the mean of the Weibull random variable in Problem 4.15 is I‘(1 + 1/B)
where F(.r) is the gamma function defined in Eq. (4.56). (h) Find the second moment and the variance of the Weibull random variable. 4.27. A voltage Xis uniformly distributed in the set {—3. —2, . . . . 3, 4}.
(21) Find the pdf and cdf of the random variable X.
(b) Find the pdf and cdf of the random variable Y = ~2X2 + 3.
(c) Find the pdf and cdf of the random variable W = cos(7rX/8).
(d) Find the pdf and cdf of the random variable Z = cos2(7rX/8).
4.28. Find the pdf and cdf of the Zipf random variable in Problem 3.70.
4.29. Let C be an event for which P[C] > 0. Show that FX(x
a cdf.
4.30. (a) In Problem 4.13, find FX(xIC) Where C : {X > 0}.
(b) Find FX(x I C) where C : {X : 0}.
4.31. (a) In Problem 4.10, find FX(x I B) where B : {hand does not stop at 3, 6,
o’clock}.
(b) Find IRA/(x BC).
4.32. In Problem 4.13,find fX(x I B) and FX(x I B) where B = {X > 0.25}.
4.33. Let X be the exponential random variable. , ' 4.50. Explain why the mean of the Cauchy random variable does not exist. (3) Find and plot 13X” I X > L). How does FX(x l X > t) diffCIflOm FXIXI? Show that E[X] does not exist for the Pareto random variable with a : 1 and xm = 1.
(b) Find and p101 fXO‘" I X > I). Verify EqS. (4.36), (4.37) and (4.38) (c) Show that PIX > r + x X > f] = p[X > x]. Explain why this is called th’ Let Y : A cos(wt) + c whereA has mean m and variance 02 and a) and c are constants.
oryless property. Find the mean and variance of Y. Compare the results to those obtained in Example 4.15. 4.34. The Pareto random variable X has edf: A limiter is shown in Fig. P42. C) satisfies the eight prope
(a) Find and plot the pdf of X.
([1) Repeat Problem 4.33 parts a and b for the Pareto random variable.
(c) What happens to P[X > t + x I X > I] as r becomes large? Interpret this re
4.35. (a) Find and plot FX(x a S X S [9). Compare FX(xIa S X S b) to FAX)
(b) Find and plot fX(x I a S X s b).
4.36. In Problem 4.6, find FR(r I R > 1) and fR(rI R > 1). FIGURE P4.2 ...
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