Queen s University Department of Mathematics and Statistics. MTHE/STAT 353 Final Examination April 16, 2016 Instructor: G.

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1 Student Number Queen s University Department of Mathematics and Statistics MTHE/STAT 353 Final Examination April Instructor: G Takahara Proctors are unable to respond to queries about the interpretation of exam questions Do your best to answer exam questions as written The candidate is urged to submit with the answer paper a clear statement of any assumptions made if doubt exists as to the interpretation of any question that requires a written answer This material is copyrighted and is for the sole use of students registered in MTHE/STAT 353 and writing this examination This material shall not be distributed or disseminated Failure to abide by these conditions is a breach of copyright and may also constitute a breach of academic integrity under the University Senates Academic Integrity Policy Statement Formulas and tables are attached An 85 inch sheet of notes both sides) is permitted Simple calculators are permitted Casio 99 red blue or gold sticker) HOWEVER do reasonable simplifications Write the answers in the space provided continue on the backs of pages if needed SHOW YOUR WORK CLEARLY Correct answers without clear work showing how you got there will not receive full marks Marks per part question are shown in brackets at the right margin Marks: Please do not write in the space below Problem [0] Problem 4 [0] Problem 2 [0] Problem 5 [0] Problem 3 [0] Problem 6 [0] Total: [60]

2 MTHE/STAT 353 Final Exam April Page 2 of 2 Let X Y Z be jointly continuous random variables with joint pdf cxyz for 0 <x<y<z< fx y z) 0 otherwise for some normalizing constant c a) Find c i c xyz yzy dydz c { so dxdydz zt dt got ) CI [3] b) Compute P Y apple ) [7] fyly ) 48y xz 48 y I z { 24 y3 E 2 y3 Then PCY E) dxdz dz Y ) y 0 < ) 2 " y3 2 and yc y 5) dy "yi) t EU Note typo : " i " should be " I " Clearly PC Y E D I

3 MTHE/STAT 353 Final Exam April Page 3 of 2 b) extra worksheet if needed)

4 MTHE/STAT 353 Final Exam April Page 4 of 2 2 Let X Y Z ) T X 0 Y 0 Z 0 ) T be 0 random vectors in R 3 that are mutually independent and discrete each with joint pmf x+y+z for x 2{0 }y 2{0 }z 2{0 } 2 px y z) 0 otherwise Wa5 W 3 } )? Let Z i be the number these vectors whose components sum to i for i 2 3 Find P Z 2Z 2 5Z 3 3) [0] Each vector Xi Yi as a multi nomial trial sum of the components 00 0 o ) 0 o ) I Tz Yi :% f f I 0 ) C 0 ) C I ) Zi IT be can regarded where the outcome is the a I # pn 3 ; 2 Then Pwiz was w3})gi3) E) 5h )3 :

5 ) 4) MTHE/STAT 353 Final Exam April Page 5 of 2 3 Let X X n be independent random variables each with an Exponential distribution with mean Let X ) and X 2) denote the first and second order statistics respectively Find Cove X ) e X 2)) [0] f x ) he e " " he " 4 x o fz k ) n n " 2 n Kye n ) 4)e i e K of n ) e " 2 e e 2) nch i ) e " l e of Xa X and Xii Xu ) z X K<)n pdfs E[e ad e K 2 e k ) " 2 o< x < k< a SoExinenkdxinnyfEhetmn4dxinnEfECexcDfrekncnDetmdkdkfEkncnilenkdxznfnIIfYnyetn4kdhnPcnDetmlK2dknnhInntnnihIn@FfexaekYoPfekeknCnheketmk2dxzdxinncnIffycnz YID ) e " " dxsdx Yaffe " " " dx

6 2) ) l MTHE/STAT 353 Final Exam April Page 6 of 2 3 extra worksheet if needed) Then Codex " im?yn# # n h h 2 h 2 h2 the ) a 2 )

7 MTHE/STAT 353 Final Exam April Page 7 of 2 4a) A triangle has vertices labelled 0 and 2 A particle moves randomly among the vertices as follows From its current location it will move to the adjacent vertex clockwise with probability p and to the adjacent vertex counterclockwise with probability p where p 2 0 ) is fixed If the particle starts at vertex 0 compute the expected number of moves until the particle returns to vertex 0 Hint: Let T 0 be the desired expected number of moves and let T i i 2 denote the expected number of moves to first reach vertex 0 starting from vertex i By conditioning set up a system of equations for T 0 T and T 2 [6] i HIP the Gditteopi on first To p i + T ) + i p ) + Tz ) T p + Tz ) + i p ) Tz p + p ) + T ) : To 3

8 MTHE/STAT 353 Final Exam April Page 8 of 2 b) Let p 2 0 ) be fixed A coin with probability of heads p is flipped until a heads is obtained Let Y denote the number of flips required The coin is then flipped Y times Let X denote the number of heads obtained in these Y flips Find the mean and variance of X [4] Condition on Y E [ X I Y ] p Y Var X I Y ) p I p ) Y ECX ) E [ p Y ) p F ) ar ) E[ p I p ) Y ) + Var p Y ) p I p ) tp + p p p + I p 2 i p )

9 h MTHE/STAT 353 Final Exam April Page 9 of 2 Xn 5a) Let Y n have an Exponentialn) distribution for n Let X be an arbitrary random variable and let X n X + Y n Show that X n converges to X almost surely [5] as Slow [ PCIXN PC { % e) n EPCY h l xl e) < a : e) E A n e X with pwb condition ee) " < convergent Geometric series ) A by the sufficient

10 MTHE/STAT 353 Final Exam April Page 0 of 2 b) Let {X n } n and {Y n } n be two sequences of random variables and suppose that X n converges to X in probability and Y n converges to Y in probability for some random variables X and Y Show that X n + Y n converges to X + Y in probability Hint: Use the triangle inequality and the fact that if a and b are positive numbers then a + b implies at least one of a /2 or b /2 must hold [5] P I Xa + Y X + y ) I e) p Xn E P l xn xl + Yn Y I e) + I Y ȲI e ) Pkixa 3 xl U { lifyl E } ) P lx xl a E) + P l Yn o o as n p YI E) So Xnt Y 4 xty

11 xl MTHE/STAT 353 Final Exam April Page of 2 6a) Let X X 2 be a sequence of random variables each having a Uniform0) distribution and let X be a random variable also with a Uniform0) distribution independent of all the X i s n! P IX c) Show that X n does not converge almost surely to X as i e) 2 to as h a Therefore Xn # X # xntx with { I probability [5] b) Let P) be a probability space Let A A 2 A 3 be a decreasing sequence of sets in and let A S n A n Let if! 2 A n if! 2 A X n!) I An!) and X!) I A!) 0 if!/2 A n 0 if!/2 A Show that X n converges to X almost surely [5] If we A then we An for So Xn at for So Xncw ) X w ) every every If WFA then there exists some m such that wet Am Then since I A w An for n h any n m } is a decreasing sequence of sets 5 Xncw ) o for all n zm So Xnlw ) Xlw ) as n p So Xn X with probability

12 MTHE/STAT 353 Final Exam April Page 2 of 2 Formula Sheet Special Distributions Beta distribution with parameters 0 and 0: 8 < + ) ) ) fx) x x) if 0 <x< : 0 otherwise E[X] + VarX) + ) 2 + +) and gives the Uniform distribution on 0 ) Gamma distribution with parameters r0 and 0: 8 r < r) fx) xr x e if x0 E[X] r VarX) r : 2 0 otherwise r M X t) for t < t r gives the Exponential distribution with mean / Geometric distribution with parameter p 2 0 ): 8 < p p) k if k 2 3 fk) : 0 otherwise M X t) pe t p)e t for t 6 ln p) E[X] p VarX) p p 2 Binomial distribution with parameters n and p 2 0 ): 8 < n k fk) pk p) n k if k 0 n E[X] np VarX) np p) : 0 otherwise M X t) p + p)e t ) n for t 2 R

I II III IV V VI VII VIII IX Total

I II III IV V VI VII VIII IX Total DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON MATH 121 - DEC 2014 CDS/Section 700 Students ONLY INSTRUCTIONS: Answer all questions, writing clearly in the space provided. If you