probpracsol

probpracsol - @@ p is a PDFJ so we new? ‘P 30 [Ii/Aida...

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H r U , * .’ i ’ f w f“. 9%} (over) ; filaflfflfigk {W 7 _ WW2} ._ ,h 5: 1:3 K “WWW;g&,,:,:mi_.imm , .,\.wwwmzw,g,,,i,w‘,-.xw,.i,_.m...,mmWWWFWHT4MW,a.,W.fi;,,¢:.k,Zs,;,:_w ( as ‘ ‘ ‘ fl ’ . wwwwwwww m l m...»,.W%._.£f‘“”mm£%_§wi'7 A <3 mmwwfiw “h“um...”we”...amupmmmzmzwuwumx. , W W w “fwmmmwwM.W.mm..mm"~mmmwm~mmmmm-mmWWW .» J T i «g a, r; kw w vvy‘ m n; V as». c537,.m m m & IP'_‘7 Mart/i 070 - l/l/r'm'é’r (QOOQ —- Fim/ 5mm 5o/07é'om5l (14 points) In the original game of Pick Up Sticks, invented by the French naturalist Buffon, a bunch of toothpicks are randomly tossed onto a tabletop that is marked with horizontal parallel lines, and the “successful” tosses are the toothpicks that land touching a line. \ / Q‘s—l Buffon, who was interested in the angles at which successful toothpicks are found to lie, dis covered that the angles m of successful toothpicks have the following probability distribution: 0 sina: if 0 S x S 7r M33) = . 0 otherw1se. Where C is a positive constant. (a) Find 0, given that h is a probability density function. 09 Sines ln {sax PDF) wus'f‘ have SMflJxr—‘l . --uo 0?) g M‘KMx “0:3 0 50 l J 11— ‘ 8 Cyraxe : COSZMXJX = C (-cosx){:r s C("CDS'IT—(—c030)) :C(l+l)=&C_ For easy reference, here is h: C’ since if U S x 5 7r M93) m . 0 OtheI'WISB, ll m x 4 M‘- ‘1‘. S X 9:; )4 Tr it Q “Srnxaix :‘MLWMS: a M=x AV: Siwxoex ._. J. T’ " 5L “(41°90 " ‘costx 0 “IT "'7 'J'Xcasx + J- ‘ a 9\JS'IJ/IX O _ __L f (-0 a) .= ~é17.(.,)+ O __ O ’- ‘0 If! E ' 10. (10 points) For this problem, use the following information about any normal (“bell—shaped” or “Gaussian”) probability density function f: 1 m/ 271' o f has the general form f m e—(e—elz/QUZ . f— f(a:)dx a .84 n+20' - f f(33)dcc m .98 ‘00 In 1959, NASA completed an exhaustive search for candidates to fly in the Mercury program, the nation’s inaugural manned space missions. Among the many requirements was that the prospective astronaut be an adult male of height between 58 and 76 inches. Suppose that, in 1959, the height of an American adult male could be modeled by a random variable with a normal distribution, with mean 70 in and standard deviation 6 in. Using the above facts and assumptions, find the proportion of adult males in 1959 who would have passed the height requirement for the astronaut program; in other words, find the prob— ability that a randomly selected American adult male had a height between 58 and 76 inches. Justify your answer by writing an integral expression that represents this probability and showing how to evaluate this integral. Since/4:70 anal G=6J na'l'i‘ce 'Han‘l' 7€= 704-6 =/u+<r' am! 5'8:- 7o—qa =/4_ar' Thus 1% F‘ahelnili'ly we Wan‘l‘ can be expneSSeal firm? graphical/y depid'ebf) qs: 76‘ f “(x— z . 1 Im- . ,H-Air 7 53 w "014vle relationship (miller! bo‘l'lq as in‘leflrqls or as mans) ,- 76‘ . 7" g 5251de = _ “6‘34" " G694" _ gmwnfi . E T?“ 55' M km‘W'HIEL ‘llm‘l'op'ffie‘l'wa areas is 0.2” J Anal 'Hre 532mm? can be, ‘30qu hr 'Hle below .' 13 19ml 'l‘lle valve). 'Flrsf use 44:3 *3 58’ °° 23: Wheat: = gal‘llxklx = l—— mat : :- oeg= no; (5mm) am we ”"° = _/_ 3 = l - = Paw-700:2 __ 5874—25 mffiflir " 7+3.- So 1% meal/Probabili‘ly we Wm .3 0.8L! —o.oa =_ ...
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