HW1 - Probability Theory and Statistics (EE/TE 3341)...

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Probability Theory and Statistics (EE/TE 3341) Homework 1 Solutions Problem Solutions : Yates and Goodman, 1.1.1 1.2.2 1.3.1 1.3.2 1.4.1 1.4.2 1.4.4 1.5.1 1.5.2 and 1.5.5 Problem 1.1.1 Solution Based on the Venn diagram M O T the answers are fairly straightforward: (a) Since T M n = φ , T and M are not mutually exclusive. (b) Every pizza is either Regular ( R ), or Tuscan ( T ). Hence R T = S so that R and T are collectively exhaustive. Thus its also (trivially) true that R T M = S . That is, R , T and M are also collectively exhaustive. (c) From the Venn diagram, T and O are mutually exclusive. In words, this means that Tuscan pizzas never have onions or pizzas with onions are never Tuscan. As an aside, “Tuscan” is a fake pizza designation; one shouldn’t conclude that people from Tuscany actually dislike onions. (d) From the Venn diagram, M T and O are mutually exclusive. Thus Gerlanda’s doesn’t make Tuscan pizza with mushrooms and onions. (e) Yes. In terms of the Venn diagram, these pizzas are in the set ( T M O ) c . 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) 1
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(c) Since Z F X A = { aaf,aff } n = φ , Z F and X A are not mutually exclusive. (d) Since Z F X A = { aaa,aaf,afa,aff,faf,fff } n = S , Z F and X A are not collectively exhaustive. (e) The event that more than one circuit is acceptable is C = { aaa,aaf,afa,faa } . (4) The event that at least two circuits fail is D = { ffa,faf,aff,fff } . (5) (f) Inspection shows that C D = φ so C and D are mutually exclusive. (g) Since
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HW1 - Probability Theory and Statistics (EE/TE 3341)...

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