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hw1soln_06 - ECE302 Spring 2006 HW1 Solutions 1 Solutions...

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ECE302 Spring 2006 HW1 Solutions January 16, 2006 1 Solutions to HW1 Note: These solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics where I thought more detail was appropriate. Problem 1.2.1 A fax transmission can take place at any of three speeds depending on the condition of the phone connection between the two fax machines. The speeds are high ( h ) at 14 , 400 b/s, medium ( m ) at 9600 b/s, and low ( l ) at 4800 b/s. In response to requests for information, a company sends either short faxes of two ( t ) pages, or long faxes of four ( f ) pages. Consider the experiment of monitoring a fax transmission and observing the transmission speed and length. An observation is a two-letter word, for example, a high-speed, two-page fax is ht . (a) What is the sample space of the experiment? (b) Let A 1 be the event “medium-speed fax.” What are the outcomes in A 1 ? (c) Let A 2 be the event “short (two-page) fax.” What are the outcomes in A 2 ? (d) Let A 3 be the event “high-speed fax or low-speed fax.” What are the outcomes in A 3 ? (e) Are A 1 , A 2 , and A 3 mutually exclusive? (f) Are A 1 , A 2 , and A 3 collectively exhaustive? Problem 1.2.1 Solution (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, hf, mt, mf, lt, lf } . (1) (b) The event that the fax is medium speed is A 1 = { mt, mf } . (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, hf, lt, lf } . (e) Since A 1 A 2 = { mt } and is not empty, A 1 , A 2 , and A 3 are not mutually exclusive. (Equivalently, since A 2 A 3 negationslash = , A 1 , A 2 , and A 3 are not mutually exclusive.) (f) Since A 1 A 2 A 3 = { ht, hf, mt, mf, lt, lf } = S, (2) the collection A 1 , A 2 , A 3 is collectively exhaustive.
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ECE302 Spring 2006 HW1 Solutions January 16, 2006 2 Problem 1.2.2 An integrated circuit factory has three machines X , Y , and Z . Test one integrated circuit produced by each machine. Either a circuit is acceptable ( a ) or it fails ( f ). An observation is a sequence of three test results corresponding to the circuits from machines X , Y , and Z , respectively. For example, aaf is the observation that the circuits from X and Y pass the test and the circuit from Z fails the test. (a) What are the elements of the sample space of this experiment? (b) What are the elements of the sets Z F = { circuit from Z fails } , X A = { circuit from X is acceptable } . (c) Are Z F and X A mutually exclusive? (d) Are Z F and X A collectively exhaustive? (e) What are the elements of the sets C = { more than one circuit acceptable } , D = { at least two circuits fail } . (f) Are C and D mutually exclusive? (g) Are C and D collectively exhaustive? Problem 1.2.2 Solution (a) The sample space of the experiment is S = { aaa, aaf, afa, faa, ffa, faf, aff, fff } . (1) (b) The event that the circuit from Z fails is Z F = { aaf, aff, faf, fff } . (2) The event that the circuit from X is acceptable is X A = { aaa, aaf, afa, aff } . (3) (c) Since Z F X A = { aaf, aff } negationslash = φ , Z F and X A are not mutually exclusive.
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