STAB52H3 Midterm Test Midterm Test October 25,2008 Duration: one hour and fifty minutes S; 0 l~ e1\t'-j

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1 STAB52H3 Midterm Test Midterm Test October 25,2008 Duration: one hour and fifty minutes S; 0 l~ e1\t'-j Please do NOT open this booklet until you are asked to do so There areq pages including this page. Please check to see if you have all the pages. Aids allowed: This test is open book. You are allowed to use a non-programmable calculator. No other aids are allowed. All your work must be presented clearly in order to get credit. Just an answer with no other work shown will only qualify for zero credit. Show your work and answer in the space provided, in ink. Pencil may be used, but then any re-grading will NOT be allowed. You will get a mark out of 90 for this test. Last name: First name: Other names: Student number: Circle your tutorial T001(Mon 9:00-10:00) T002(Mon 13:00-14:00) T003(Mon 15:00-16:00) Page 1 of9

2 1) For two events A and B, it is known that pea) =!, PCB)=1.. and P(AnBC) =1.., 439 [5 points] a) Calculate pea u B). p (A-v f?:,') ~ pon + (?r IS)- PClt-B) /' ;::-l-+j - ( '.-~ ) 4 3 '~ ~ - p (AI- P (I?::,')-1-r (~ ) ~ :::- f7 (A-n ~c) ~ () (ts "') [5 points] b) Calculate P(ACIB), - ~ -tj ~ ~ P(ItC g) P(~' ) ~(1tB )1-r (A fsc) ~ pefr ) ~) P(It-&) ~ 1,-.~ 't ~ >' ,- --- q Page2 of9

3 [6 points] 2) Suppose that A and B are events such that P(AIB)= P(BIA) and P(AuB) = 1 and pea n B) > O.Prove that pea) > 0.5. Sol P(AIR) = P(RIA) <=>P(AnB) = perna) =>P(A)=P(B) PCB) pea) =>1= P(A u B) = 2P(A) - pea n B) =>P(A n B) =2P(A) -1 P(AnR) > O=>2P(A)-1 >O=> pea) >0.5 [10 points] 3) Suppose there are three boxes each having five balls as described below. Box-l contains five white balls. Box-2 contains five black balls. Box-3 contains three white balls and two black balls. A box is picked at random and then three balls are picked at random (without replacement) from the box selected. Calculate the probability that at least two of these three balls will be white? Sol: Denoting the event 'at lest 2w' by 2w. By thm oftotal probability, P(2w) = P(2wlboxl) P(Boxl) + P(2wlbox2) P(Box2)+ P(2wlbox3) P(Box3) 6+1 = 7 ANS/10 = 0.7 ANS/3 = I.(1/3)+0.(1/3)Ja~t\a~L! ~ ~ 3 ~ ::-50 g. Page 3 of9

4 ANS+(1/3) = ) LetXbe a random variable with the following probability density function: I(x) CX2 = { ~(3-X) O<x::::;;2 2 < x::::;;3 otherwise [3 points] c) Calculate P(1< X < 2.5) F (2-.r ) -- \-- C t). fj --L~ [! --(5-25 -)J (~ (~ o ~2-:S~ ~, I~ /1 Page 4 of9 -.."... ~ --/"

5 [6 points] 5) The random X has a uniform distribution on the interval [0, 10]. Calculatep( X + ~ >7). Sol This is question 17 p 250 SOA p( X + ~ >7 )=P(X2-7X +1O>O)=P((X -5)(X -2»0) = P(X>5)+P(X <2)=-+-= [6 points] 6) LetXbe a random variable with the following probability density function: O:$x :$1 otherwise' Find the probability density function of Y =5_'!. :: h ('f ) 2 Sol This is question 6.29 p302 Wackerly and Mendenhall. Solon p 143 Sol manual. 'L 1 <j L 2ls -I) J 1- ~ (S'- ji ( :;:A- <5 \ 12-- ' i ~ () ~1 ) -t \f- t~ -'1) ~ ~ 0 Page 5 of9

6 7) Let X and Y be continuous random variables with joint density function evj"), x = cxy for O<x<l and O~y~l.. { o erwise fx,y(,y) 0 th [5 points] a) Calculate the value of c. \ ' fsc?t(~/) ~~ ~ o 0 c f~ J C) D ~ L ~ J [ r ~ lj \ ' (r r'1 ) tr, ~ ) ~ ) [5 points] b) Calculate P (y2 < X) 2- \/\ YL Y\,' / l hi /\I - 7( \ '\ J Ilf: - r - {~ Page 6of9

7 8) Let X and Y be continuous random variables with joint density function 24XY for 0 < x < 1and 0 ~ y ~ 1- x f x y( x,y) =.., { 0 0the~se 1 [5 points] a) Derive the marginal density of X...~ I-? 1('" yo J~<XJ j ~~')\ ~, lor \ 'L J ';- ~'K') ~ :l-\.\ 0\- l1-0 ~ v ;::: I:L?<- (\ ~ 7\ ('i ) (!)~./'-. J;.., [2 points] b) Derive the conditional density of Y given X =x. r C'")'1L1) 30 Y f' 1) 'f.. 'Y\\,- \.~ / 1'--aC' Sol This is question 28 p 243 SOA Exam May 2003 \ t Page 70f9

8 9) Let X and Y be discrete random variables with joint probability mass function: Determine AX,y>={j; ) if x = 1,2, y = 1,2 otherwise. [6 points] a) the value of the constant k '2-2-k [7 points]b) P(X > I I Y =1) S &/ f. r..j)..c:/\v' r- ry-- 'vv- - ~,') ( X '/ 1 (\ "'1 ~ \ ) pcy;::-\) '\ <pl ~ :5 ~. -" - -~.3 Page 80f9

9 [6 points] 10). Suppose that X ~ bin(2, Calculate P( X + Y = 4) 3/4), Y ~ bin(3, 1/2) and X ad Yare independent p( X -+ Y :=- tr J:::' P( >U-1t1 j ~ 2-) --t P (>< '" I) '! "'- s ) = pc'f ~ 2- ') P tf",2- ') 1: P(I< ~I) Pr;;o3,) c-' X.L~ ~(~)Gt(~)~(~X~)(iJ\ -\ ( 2 ' ~ ( 1\6 ~ \ 1\ ( '>\ L ) t\i-) (if') ~ 3 \ 2.-) ~J ~ - / Page 9 of9

UNIVERSITY OF TORONTO SCARBOROUGH Department of Computer and Mathematical Sciences Midterm Test, March 2014

UNIVERSITY OF TORONTO SCARBOROUGH Department of Computer and Mathematical Sciences Midterm Test, March 2014 STAD57H3 Time Series Analysis Duration: One hour and fifty minutes Last Name: First Name: Student