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hw2_soln - Probability and Stochastic Processes A Friendly...

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Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J. Goodman Problem Solutions : Yates and Goodman,1.5.2 1.5.3 1.5.5 1.6.1 1.6.4 1.6.7 1.7.4 1.8.3 1.8.4 and 1.9.1 Problem 1.5.2 Let s i denote the outcome that the roll is i . So, for 1 i 6, R i s i . Similarly, G j s j 1 s 6 . (a) Since G 1 s 2 s 3 s 4 s 5 s 6 and all outcomes have probability 1 6, P G 1 5 6. The event R 3 G 1 s 3 and P R 3 G 1 1 6 so that P R 3 G 1 P R 3 G 1 P G 1 1 5 (b) The conditional probability that 6 is rolled given that the roll is greater than 3 is P R 6 G 3 P R 6 G 3 P G 3 P s 6 P s 4 s 5 s 6 1 6 3 6 (c) The event E that the roll is even is E s 2 s 4 s 6 and has probability 3 6. The joint proba- bility of G 3 and E is P G 3 E P s 4 s 6 1 3 The conditional probabilities of G 3 given E is P G 3 E P G 3 E P E 1 3 1 2 2 3 (d) The conditional probability that the roll is even given that it’s greater than 3 is P E G 3 P EG 3 P G 3 1 3 1 2 2 3 Problem 1.5.3 Since the 2 of clubs is an even numbered card, C 2 E so that P C 2 E P C 2 1 3. Since P E 2 3, P C 2 E P C 2 E P E 1 3 2 3 1 2 The probability that an even numbered card is picked given that the 2 is picked is P E C 2 P C 2 E P C 2 1 3 1 3 1 1
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Problem 1.5.5 written i jk where the first card drawn is i , the second is j and the third is k . The sample space is S 234 243 324 342 423 432 and each of the six outcomes has probability 1 6. The events E 1 E 2 E 3 O 1 O 2 O 3 are E 1 234 243 423 432 O 1 324 342 E 2 243 324 342 423 O 2 234 432 E 3 234 324 342 432 O 3 243 423 (a) The conditional probability the second card is even given that the first card is even is P E 2 E 1 P E 2 E 1 P E 1 P 243 423 P 234 243 423 432 2 6 4 6 1 2 (b) The conditional probability the first card is even given that the second card is even is P E 1 E 2 P E 1 E 2
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