2008_Fall_MT_1

# 2008_Fall_MT_1 - I Time 18:00-19:40 b‘WL‘L 66M ML Date...

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Unformatted text preview: I Time: 18:00-19:40 b‘WL‘L 66M ML,- Date: October 24, 2008 Instructor: Dilek Gtivenc / MATH 230 I ,TERNI EXAM I . I, _ IMPORTANT 1 Check that there are 4 questions in y ur be (let. 2 Do NOT use your rnobﬂe‘ phone as a alc ator. T m it off during the exam. 3 Show an your work. Correct results Without su 1cient eiplanation and correct notation might not get full credit. /’ 4 Write your name on ea h page. 1. Suppose that there are N +1 boxes. Box numbered k contains k red N- k white balls, ( k= 0, 1, 2, N) A box is chosen at random and n drawings are made with replacement. Deﬁne events A ={A-11n balls turn out to be red}, B = { The (n+1)sz‘ drawyiel‘d's a red ball} Find ; , a) P(A| Box k is chosen) (k= 0, 1,2, N), (6 points) b) HA), ( 7 points) 0) P(A n B), (7 pOints ) d) P(B]A). (5 points) a) {CS LAOSM):W«—-7Z'(ggxk_fa5 Chaim} I m . cum aniszeé; U stir") : WWW—"M /«‘/L: <3 “M‘- ': 7"; r N fi/ n A/ 1,} {2.5.4 , a» P(BO/M)+ ﬂag/ﬂ it) e —- 7%? (EN 2"? ft ) — FKﬁWWA I/BOHWEWM'13nrw ~ +W€~>HM8N> H - o M K .e-v / o w—w-‘I _...-.—-—-' \ .—'-—-'...w—»— ‘r-an“ "—"“ .n- 2“ F m J "J" NW N”- "{ WWI N 2 :1 / (M M (“’74 {tin /’ my \ i E “’4” W ‘( 4/'9.;4///M—/3Wé/~3ﬂ"“°/ . :2) MM 6) : J, ( 0 7e {WW—re ...,.._..) a (. j W. aesesve ‘- ==~> l! / [ 1 7L inf—1+ f/Vmw’ / 7L 2.” + u “IL/V”? RESERVE NAME: ........................ .. 2. Suppose that you have two coins, one balanced and the other with probability p of a head. You select one of the coins at random and toss it twice. Determine the probability that a) the ﬁrst toss is a head, (6 points) b) the second toss is a head, (6 points) c) both tosses are heads. (6 points) ‘ (1) Show that events, “ﬁrst toss is a head" and “second toss is a head” are independent when only, if the second coin is also balanced. (7 points) head" ff! ﬁll/JG #ﬂfﬁﬁé was! a! w'} 5 ﬂats/tsunami 001277 F: If; I FM): WPNQMIFDHWFQ PCs/ﬂ) / ,L g ﬂ I I Zip-{Zip 4 Z " f/W/f)’/0 ‘- -’ I. :: .— _. gamut/£2— ?(Hltg)=,’%.%%_é_ pﬂq/F) F rain (’7’?) o.) ﬂ:§in’:/- 74.75: is a C) ME" I I glFWer’FU/jﬁfﬁngtF-f) 5’ An mm)— ﬁfe/{g Maggi; __. Z— 9' 87 47L L , a) 3, ,7..- T— ‘4; f t f __ . / .— __ vi:- .— {J _,___—-- ...——"——-—___. _._/ .a / .— axons)! C/‘WJ’J [:5 “J50 5? é’GVLé’wﬁao/(ﬂﬂ? NAME: ........................ .. 3. a) Find the value of k such that ( )_ (x+1)k x =1,2,3,5 g x n 0 otherwise is a probability distribution function of random variable X. (6 points) b) Find P( X> 2) (5 points) c) Find expected value of X. (6 points) (1) Find emulative distribution function of X and graph the function. (8 points) a) mob,»— (nomad/22+ (5mg 7. g /5é=£ 52;; £575; WWW 0000: [Kerri/3,? jg) ﬂabaé :P(X=33f/957<¢S>= 74; 7:- 7? a %=%... 3 3 ‘4 .LS" é _ 56' . 2,_ ____ _ ,_____. Hwy"...— (3) £00: 73%“2’51 #5 i /5 i5 Q00 I} a!) 6, x4: 1 J } i—mt 2— 15>“; 19 6m:— x as ' 5' léX‘i-g: /S f L h 7 9 _r 55x45 ’3. /S 2 . J: /; 6 _.__u‘5-[.=:=@ , _____.,.__..____._.;:__n, K i a 3 5 / NAME: ................ . . 4. Suppose that there are :1 components in a system and they operate independently of each other. System fails if two or more components fail. For a single component, if, the number of periods until failure of the component (including the period of the failure), Y has a geometric distribution with parameter p, a) Find the probability that failure of the system will occur after the k-rh period.(15 points) b) What is the expected number of operating components at the end of k periods? (5 points) I: a) #5 pf payaer w‘trlz’zf a 5/26/11 mac/whet ﬂax/5‘- 31’”! yn/ @wmé’xtzﬂ’fﬁ (/9) #) affij=z (p €56.21 ___ lemme}: PKYakwﬁik {at “F Velma; 7"+——-J {34: n l I ﬂ -« k ’/ _._._ é Irv—i, 5x {pry-3.3.,n ,‘f’ 5 O’c’fwrS E/Jg /-f/—hc( (f L‘s [5% f _ ,5 X: # 0&4! com/004m}: amid é fay/“bag gig/341’“: ’él/Za \ » 2? ﬂ—X XXV gr‘nK/J‘JCﬁ/é) d/(X)=(Q)K?£) (/—-?é) Xsaj/{nﬁ/L‘ p(X>”-2>=/O(XM»GPram.) ,3 L” fir-19W?” i“ W 7-!- ( (ft) //— at; * iv 0"" ,ét’lﬁ at. s— d a . :zﬂfﬂ—f' ,— a to) 1:00;”? RFSERVF ...
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## This note was uploaded on 02/01/2010 for the course CS MATH-230 taught by Professor Dilekgüvenç during the Fall '10 term at Bilkent University.

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2008_Fall_MT_1 - I Time 18:00-19:40 b‘WL‘L 66M ML Date...

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