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STOR MATH 634 Graduate Probability I Comprehensive Written Exam FILL OUT THE INFORMATION IN THE BOX August 2021 Name READ THE FOLLOWING INFORMATION This is a closed book and notes exam There are 4 questions Attempt all questions You may appeal to any result proved in class without proof unless you are speci cally asked to Give a complete proof All questions are worth the same total amount 15 pts Breathe and try to relax and think about each problem slowly Remember I am testing not just if you can get the nal correct answer but your technique in approaching problems Partial credit will be given even if you are not able to completely solve a problem but make headway in solving the problem STOR 634 CWE 2020 2021 August 2021 15 Q1 Suppose F P is a probability space and Fk k cid 62 1 are a sequence of sub elds of F Show that the sequence Fk k cid 62 1 is independent if and only if each of the pairs F1 F2 Fn Fn 1 is independent for n 1 2 1 STOR 634 CWE 2020 2021 August 2021 15 Q2 Let Xn n cid 62 1 be a sequence of non negative random variables all de ned on the same space and let mn n cid 62 1 be a sequence of strictly positive constants with mn For each xed n cid 62 1 suppose the probability density function of Xn is given by Note this implies for any t cid 62 0 that P Xn cid 62 t exp mnt De ne the random variable fn y mn exp mny y cid 62 0 Y sup n cid 62 1 Xn a Suppose mn log n 1 for all n cid 62 1 Show that b Suppose mn log log n 4 for n cid 62 1 the n 4 is just to ensure that mn 0 and increasing to understand the problem it is easier to just thinking of the above as the sequence log log n Further assume Xn n cid 62 1 are independent collection of random variables Show that P Y 1 P Y 1 10 5 2 STOR 634 CWE 2020 2021 August 2021 3 STOR 634 CWE 2020 2021 August 2021 15 Q3 Suppose F is a general nite measure space and let fn n cid 62 1 be measurable collection of functions Suppose g cid 62 0 is another measurable and integrable function such that fn cid 54 g for all Consider the function f lim sup n fn Show that f is integrable and further cid 90 cid 90 f d cid 62 lim sup n fnd 4 STOR 634 CWE 2020 2021 August 2021 15 Q4 Fix 0 and Suppose Xn n cid 62 1 is a sequence of independent Bernoulli random variables such that for each n cid 62 1 Xn is a Bernoulli 1 n random variable i e for each n cid 62 1 P Xn 1 1 n P Xn 0 1 1 n n cid 88 i 1 De ne Sn Xi 5 5 5 a Show that if 1 then Sn converges a s to a nite random variable b Show that if cid 54 1 then Sn a s c Show that in the setting of part 4b Sn E Sn a e 1 Hint for part c this is similar to the nal in STOR 634 in Fall 2020 Precisely De ne nk inf n 1 E Sn k2 Let Tk Snk Show that Tk E Tk 1 a s by arguing that for every xed 0 cid 88 k 1 P Tk E Tk E Tk Try to use this to complete the proof of c 5 STOR 634 CWE 2020 2021 August 2021 6


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UNC-Chapel Hill STOR 634 - Comprehensive Written Exam

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