Exam II Solutions

Probability and Statistics for Engineering and the Sciences (with Student Suite Online)

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Unformatted text preview: Sum/Km. $001- £xnm IE 1. (30 pts.) Each bit in a string of a 20-bit message is sent 3 times over a noisy communications network. Suppose that bits are corrupted (i.e., changed from Is to OS and from OS to Is) independently with chance p = 0.10. A majority detector is used at the receiving end to infer the bit sent. In particular, for each bit sent . . . . . . if 2 or 3 of the bits received are 0s, call the bit sent a 0 . . if 2 or 3 of the bits received are 1s, call the bit sent a 1 a. For a particular message bit that is sent 3 times by the sender, what is the chance that the receiver correctly infers this bit? b. What is the chance that the entire 20-bit message is correctly inferred? c. How many corrupted bits do we expect the receiver to infer for the 20-bit message? d. What is the chance that 18 or more bits in the 20-bit message are correctly inferred? @ ‘F( BECEIVEK— Conn-(5‘11“; I'JFEVLS WV") V( 2- 0“ 3 “fin—fr- 11.x magmas) P( 1—- (mutant Mean/(oh Pf} (om-IL? "(can/£0) .3 b S (i)(.‘7a)7-(ow) +- (g)(79) (do) : ,7—1'} 4- .77-‘1‘ / ((1; 4m”; man“) 4— WP? (mm-r M‘fin-‘OD‘F V010 (amour NM) 1 I? 1. 7-0 a a (I ;)(.771_) (:02?) 4' (7'0 .777.) 76022) +(‘33X3729 (-9289 422D (“WW”) 3 .0373? +— .}2£47 + ,5:::: 3: flip}? 2. (20 pts.) This problem concerns the tossing of a pair of fair six-sided dice. The expression ‘snake-eyes’ refers to the event of having both dice show one dot. a. What is the chance of snake-eyes on any given roll of the pair of dice? b. What is the chance snake—eyes first appear on the 10tll toss of the pair of dice? c. What is the chance that it takes at least 4 tosses of the pair of dice to obtain snake-eyes? For maximal credit, give an answer involving a finite number of terms. d. How many tosses of the pair of dice should you expect before getting snake-eyes? @ 43; 6) P( sphlte if!) var accents 7’55 lo) = C;- 3i?) (L 3‘) ’5 .021557 (9 P( Ar (AS/«bf + 11935;. Fan suit; are; y: /- V! Arrf Mn" 2(7)”; Fm 5mm; Eve's) /- 23707199: er;- 219252: all. 3115324)] t [— [Kc/72:51) 4. P(7—4'9962)* Pl: 791551)] N :. .. ,L 3 35 i 96 ’f’ i ’L + (:52 .7/7 36 7é @ ,M:-J-—:' J”=Bé ” (at) 3. (20 pts.) 1m pi = Chance that the i"1 component works Suppose that the various components in this problem operate independently. a. (A series system - works only if all components work) Determine the chance that the system below works - express your answers in terms of p1, p2 , p3 Wit/mam mag, Ft {awe n “was n 3 (James) I“? V/lwov-ks) Pé? warts) V6 mag-Ia) 5 [PI Ft F3 ( b. (A parallel system - works if at least one 0 components work) Determine the chance that the system below works - express your answers in terms of p4, p5, p6 M :- {—- [’(f HMS) P5 Fan-J) P/4 Phil—s) (Problem continues on the next pay; c. The previous two systems of parts a. and b. are placed in parallel. Determine the chance this new system works — in terms of p1,p2,p3,p‘,p5,p6. W llfiwfi 53371504 was) = Vt dumps U 9 ”mt-J) NW” am a way-k5) + m we) —-PCéuark.s n awe) ML . PI im“ = PM +W’ M .3 .— C {(MJE’I- (I-r+)( 11M 11)] P( [219966. 57375“ wot/2;) ’ 1!- “mews 5Y5TEM Fm“) 2 =(' VK4FMI~S ()bFAn-s) if WW“; 1—- P(a Fan‘s) Pa. Fm») _ Z _; ' ~ (I’ PIF201LI all-:1 l’rfl/v-gfll’nfl 7» -_ (—— D‘Prfirflfll-VW-nxl-WJJv 4. (19 pts.) In the game of Chuck-A-Luck you, the contestant, pick a number from 1 through 6. Then the operator of the game throws three dice. For a $1 bet, if all three match the number you chose, you win $3 from the operator. If there are two matches you win $2, if there is one match you win $1, and if there are no matches you lose $1. The probability mass function for net winnings follows: x P(Net Winnings = x) 3 1/216 2 15/216 1 75/216 -1 125/216 a. For a single $1 bet, find the mean net winnings, u. A >,,J;)+ 2/5.» (<5;- + -: (5)6!) b. For 1‘ single $1 bet, find the standard deviation of net winnings, o ‘- E ,_ (fl! fl@1(7‘")+ 7’ (2/4) 4' [194W (772)] £5)?” 3:12:17. '9 rzl’la‘ @ c. Suppose you place n $1 bets' m Chuck-A-Luck with net winnings x1, x2... .,x. As n gets big, what’ s the approximate value of x? 76—h“ fie fist-14 Reluwl’ .—$.’Z / z“ A d. Suppose you place n $1 bets in Chuck-A—Luck with net winnings x1,x2,...,x_. As n gets big, what’s the " .- 70 7 f approximate value of s— — ’—-—-E(x,— x)’? n - i=1 5»? r/ 50 5 theta I" abw‘l" $/.// 5. (10 pts.) The average February temperature in Rapid City is recorded .for the years 1888 through 1985 in the publication Rapid City Climate]. For these 98 data points the average, i, is 22.3 degrees Fahrenheit and the standard deviation, s, is 7.1 degrees Fahrenheit. Recall that °c=.5_° °F_.1_6_0. , _./‘° 9 9 0: 71+ 7 6. (6 pts.) Terminology. g a. If AnB = Q, then we say that A,B are J72 l b ”‘k events. 0 / _ b. If PM A B) = P(A)P(B), then we say that A,B are M events. 1Miller, J. (1986), Rapid City Climate, Institute of Atmospheric Sciences, SDSMT, p. 19. ...
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