Solutions Manual to Probability Random Variables and Stochastic Processes

S. Unnikrishna Pillai......Page 1
Objective......Page 846
ERRATA & COMMENTS ON THE TEXT......Page 847
1) Introduction......Page 853
2-16 M, B Problems 2-16 through 2-19 and 2-21 are problems in combinatorics that the Text has not.........Page 855
3-4 Write down the binomial theorem expansions of and , then add them together.......Page 856
3-11 D, F The concept of stakes may not be clear. What is meant is that, on each play, bets an am.........Page 857
4-13 Equations which were in the Third Edition of the Text, but have been removed, are for the de.........Page 858
4-26 F, M First fix the typo: The system should have 1000 components, not 100. Next, note that Eq.........Page 859
5-8 Use Eqs. 5-16, 4-30, and 4-44.......Page 860
5-25 I Use Eq. 4-56 for the binomial random variable. (a) Use Eq. 5-46 for a direct approach, whi.........Page 861
5-38 This is a workhorse problem. (a) F Note the typo: The characteristic function should read . .........Page 862
5-50 F The description of the experiment is confusing. What is the length of the run? The problem.........Page 863
6-3 Begin with a diagram of the region of integration—the graphical approach.......Page 864
6-20 (a) Use Eqs. 5-18 and 6-43. (b) Use Eq. 6-54. (c) Use Eq. 6-60. (d) Use the equation followi.........Page 865
6-32 (a) Let . Use a graphical approach to develop an iterated integral expression for , as in Ex.........Page 866
6-45 (a) Define and . Find the joint p.m.f. of and , then find the two marginal p.m.f.s from the .........Page 867
6-58 All basic stuff.......Page 868
6-74 Use Bayes’ theorem, Eq. 2-44.......Page 869
7-9 Use . Use Problem 6-51a to limit . The fact that is not needed.......Page 870
7-21 I, D No conceptual difficulties here, but this is a tedious problem. First deduce that is un.........Page 871
7-27 Use the Cauchy criterion, Eq. 7-117. Define . Show that converges, and use this fact to show.........Page 872
9-7 (a) Use inequality 5-89. (b) Let , . Find the region of the -plane where , and integrate the .........Page 873
9-11 B, I, M The Text does not fairly prepare you to solve this problem. In the discussion of lin.........Page 874
9-23 F You must assume here that is WSS (or, equivalently, that has a constant mean). The discuss.........Page 875
9-32 (a) Use the solution to Problem 9-29, or otherwise find the impulse response function. Use E.........Page 876
9-41 B To solve this problem, it is best to use the general convolution theorem. There are two ve.........Page 877
9-47 Use inequality 9-181 to show that for . Deduce that .......Page 878
10-7 Use Campbell’s theorem, Eq. 10-102. Note that can be zero only if no Poisson points arrive i.........Page 879
10-17 See hint 10-16 above.......Page 880
10-25 M Find the impulse response function (see Example 10-2). From the deterministic input, , fi.........Page 881
11-1 D Finding the whitening filter is easy, especially if you note that . Finding the autocorrel.........Page 882
11-8 Write as a double integral. Transform to a single integral as in Eq. 9- 59. Differentiate wi.........Page 883
12-2 I, D Note that is WSS. Use Eq. 12-35 to deduce that as . Find both sides of this limit expli.........Page 884
12-7 I, D! Assume that for some and all . Use Eq. 9-37 to get . Change variables to , . Break up .........Page 885