soln1 - Probability and Stochastic Processes Homework 1 and...

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Probability and Stochastic Processes Homework 1 and 2 Solutions October 4, 2001 Problem Solutions : Yates and Goodman,1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.3.1 1.3.2 1.4.1 1.4.2 and 1.4.3 Please contact the TA if you have any questions. Problem 1.2.1 (a) An outcome specifies whether the fax is high ( h ) , medium ( m ) , or low ( l ) speed, and whether the fax has two ( t ) pages or four ( f ) pages. The sample space is S = { ht , h f , mt , m f , lt , l f } (b) The event that the fax is medium speed is A 1 = { mt , m f } . (c) The event that a fax has two pages is A 2 = { ht , mt , lt } . (d) The event that a fax is either high speed or low speed is A 3 = { ht , h f , lt , l f } . (e) Since A 1 A 2 = { mt } and is not empty, A 1 , A 2 , and A 3 are not mutually exclusive. (f) Since A 1 A 2 A 3 = { ht , h f , mt , m f , lt , l f } = S , the collection A 1 , A 2 , A 3 is collectively exhaustive. Problem 1.2.2 (a) The sample space of the experiment is S = { aaa , aa f , a fa , faa , f fa , fa f , a f f , f f f } (b) The event that the circuit from Z fails is Z F = { aa f , a f f , fa f , f f f } The event that the circuit from X is acceptable is X A = { aaa , aa f , a fa , a f f } (c) Since Z F X A = { aa f , a f f } 6 = φ , Z F and X A are not mutually exclusive. (d) Since Z F X A = { aaa , aa f , a fa , a f f , fa f , f f f } 6 = S , Z F and X A are not collectively exhaus- tive. 1
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(e) The event that more than one circuit is acceptable is C = { aaa , aa f , a fa , faa } The event that at least two circuits fail is D = { f fa , fa f , a f f , f f f } (f) Inspection shows that C D = φ so C and D are mutually exclusive.
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