com03sB-or - DOCTORAL QUALIFYING EXAMINATION SPRING 2003...

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Unformatted text preview: DOCTORAL QUALIFYING EXAMINATION SPRING 2003 Probability S lv any three pr iems for a full credit. Indicate below LLECTLE which three problems you have attempted by Circling the appropriate numbers: 1 2 3 4 NAME: SOLAR ID#: Start your answer on each question sheet. Attach all ex sheets you use to the appropriate sheet. Hand in all qu sheets. ([1 r!- H U1 rt 91 ._I O :1 Late January 23, ?J03 Time Of Exam: 1*2PM 1 Blade of Ex-m: Stony Brook Union Auditorium ID#: Each day an experimental animal is exposed to a certain set of stimuli designed lo elicit a particular response. Let AR be the event that the animal makes the desired response on the kth day: At I he the event that the animal does not make the desired response on the kth day, and suppose that RAM | At) ; 18 and Pia/1m 5 ‘4k It} : a, where 0 C‘ a q 18 31' Let Pt : P(f1k):1_P(‘4kl)‘ a. Show that pm Z CZ + (/3 _ 05H3k- b. If ’62] and p1 30= show that Pt =1—0—alk—1- a ' (p- ) = —. C. Show that Iii—13:] k 1+0 —- [3 Il)#: 2. A tea—drinking lady starts out with two boxes, each of which contains n teabags. Each time she needs a cup of tea she selects one of the two boxes at random and takes a teabag from it. What is the probability that the {m k)th teabag will empty one ofthe boxes: where 0 S I: <; n? ID#: ,1 J. Let X, Y, and 2 be independent random variables all of which have the exponential density f(x) = é’xax > 0- Find the probability ofthe simultaneous occurrence ofthe events that X E I" and X E 22'. ID#: 4‘ Let X and J" be independent random variables with means both zero and common 2' _ '3 , 1 2 variance ‘17 - L61 2 _ ‘X + Y' Fmd 12(2 IX =30, ...
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