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part3

# part3 - ECE 863 Analysis of Stochastic Systems Fall 2001...

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1 ECE 863 – Analysis of Stochastic Systems Fall 2001 Solutions for Homework Set #1 Michigan State University Department of Electrical & Computer Engineering Problem 1 2.4 (a) The sample space S = set of ordered pairs which can be expressed as: (i, 1) (i, 2) …… (i, i), for i = 1, 2, 3, 4, 5, 6 Therefore: S 1, 1 2,1 2,2 3,1 3,2 3,3 6,1 6,2 6,3 6,4 6,5 6,6 = b g m b g b g b g b g b g b g b g b g b g b g b gr , , , , , , , , 2.4 (b) 4,1 , 4,2 , 4,3 , 4,4 b g b g b g b g m r 2.4 (c) 3,3 , 4,3 , 5,3 , 6,3 b g b g b g b g m r 2.4 (d) 6,6 b g m r

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2 Problem 2 2.5 (a) Each testing of a pen has two possible outcomes: “pen is good” (g) or “pen is bad” (b). The experiment consists of testing the pens until a good pen is found. Therefore, each outcome of the experiment consists of a string of “b”s ended by a “g.” Assuming that each tested pen is not put back in the drawer, the sample space S g, bg, bbg, bbbg = l q . 2.5 (b) In this case, we simply record the number of pens that have been tested. Therefore, S 1, 2, 3, 4 = l q . 2.5 (c) In this case, S gg, bgg, gbg, bbgg, bgbg, gbbg, bbbgg, bbgbg, bgbbg, gbbbg = l q 2.5 (d) S 2, 3, 4, 5 = l q Problem 3 2.9 A 5, = b g lifetime is greater than 5 B 7, = b g lifetime is greater than 7 C 0, 3 = b g lifetime is not greater than 3 A B 7, « = b g lifetime is greater than 5 and 7 A C « = ∆ lifetime is greater than 5 and not greater than 3 A B 5, » = b g lifetime is greater than 5 or 7
3 Problem 4 2.13 (a) A B C A B C A B C c c c c c c « « » « « » « « d i d i d i 2.13 (b) A B C A B C A B C c c c « « » « « » « « d i d i d i 2.13 (c) A B C » » 2.13 (d) The events in b above A B C b g n s b g » « « 2.13 (e) A B C c c c « « Problem 5 2.18 Since the event a,c l q can be decomposed into two mutually exclusive events: a l q and c l q , fi = + = P a,c P a P c 5/8

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part3 - ECE 863 Analysis of Stochastic Systems Fall 2001...

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