Points Recall Bayes’ Thm: Let B 1 , , B n be a fi nite...

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Probability Mid-Term Exam Feb 26, 2013 Justify all reasoning. No books, notes or calculators. Five problems; each worth 20 points. Note second part of Problem 5 is on second page. 1. (20 points) Recall Bayes’ Thm: Let B 1 , ..., B n be a fi nite family of mutu- ally exclusive and exhaustive events and let A be any event. Then P ( B k | A ) = P ( B k ) P ( A | B k ) P i P ( B i ) P ( A | B i ) . Use Bayes’ Thm to explain the following by de fi ning appropriate events and some sample numbers. A group of people (e.g., engineering students) may have a characteristic (e.g., dressing conservatively). Under what conditions does one make a large, quantitative error by deducing that someone with the characteristic is actually a member of that group? 2. (20 points) Suppose that R is the time of arrival of the fi rst customer to a store. The density of R is given by f R ( x ) = ½ λe λx if x 0 0 if x < 0 ( a ) Find the probability that the fi rst person arrives between time x = 3 and x = 5 . ( b ) Find the variance in arrival time of the fi rst customer.

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