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Filled-in practice 2

Filled-in practice 2 - Stat400 [email protected]" 12nd 2011...

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Unformatted text preview: Stat400 Lecture @131" 12nd,, 2011 Page 1 Practice problems 1. Heidi and Alex have agreed to meet for lunch between noon (0:00 pm.) and 1:00 pm. Denote Heidi’s ar— rival time by X, Alex’s by Y, and suppose X and Y are independent with probability density functions 3\$20§\$§1 _ 5y40§y§1 fX(\$> _ l 0 otherwise fY<y) _ l 0 otherwise (a) Write doyvn the joint p.d.f. of (X, Y). I, / ondepwdwt >0, ' JEN/w : fx(xlfl/(<(l> : “1:74, Ogre/pew (b) Find the probability that Heidi arrives before Alex. W X < X l 3 5 7c yyyy @an ( felt: -. O Man—ram V . ..V_s.,_ Stat-400 Lecture 22, Apr 12nd, 2011 Page 3 3. Suppose that commercial airplane crashes in a certain country occur at the rate of 2.5 per year. Assume that such crashes follow a Poisson process, (a) Starting from now, let X denotes the time un— til the next crash, what is the distribution of X? Write down the p.d.f. of X. rate. A .: 2‘ g- Cmf/iesnu I Xm Iii/NW)? : 7V "04 yEW/mxﬁ' fol”) C blgé Q“ /. an (b) What is the probabilitythat the next crash will occ u r wit h i n 5th.,reewmont,hsl) Brno/47M; :1 U'Mjéd/l/ ﬂair. Wm 02\$») :: F(0,2§~):/-Q ﬁg 0‘25; Stat400 Lecture 22, Apr 12nd,, 2011 Page 2 2. According to an airline industry report, roughly ,1 piece of luggage out of every 200 that are checked is lost. Suppose that a frequent—flying businesswoman will be checking 120 bags over the course of the next year. Using Poisson approximation to find the prob— ability that she will lose 2 or more pieces of luggage. X f 4:6 luggage am: We? /20 ﬁgs x” [9020/ Rx») : l~l>(X<l) 3:5) *9: Parka/l <05) Q l a [fr 0/ /» iggmm‘mzaﬂ Porgy)” Fm gag 3 / Wan wave/We #aW/zﬂ » a ~ . rrrrr w»! ’ I I (9pm I MILK/n M ' “ * I X 1% 6 ” 00/7/15 W1€(§/JQCQ)/ X/v R9159”! M w marge #MW%€T M27739 mm to Oak/me Mt amt (/A/EXFa?) 0g (3? WUMa magi??? 750 (9526/08 (Xi/L WK Stat-400 Lecture 22, Apr 12nd7 2011 Page «1 4.(a) The tensile strength X of paper has ,u :55; and a =A§‘(pounglsmper square inch). A random sam— ple of sizeGi = 10ms taken from the distribution . W I u u of tensrle strengths.w Compute the probability that the sample mean is greater than 29.4 pounds per square inch. (Hint: use Central limit theorem) Stat400 Lecture 22., Apr 12nd; 2011 Page 5 (b) Let X equal the number out of n : 48 mature aster seeds that will germinate when p = 0.75 is the probability that a particular seed germinates. Approximate P(35 g X g 40) using normal dis— tribution. (MW/f, rim—pr ><m M45? 075“) RJ/A/(sa 7) :nprrja) Wgysx :40) : Flegtof pwqjj (/ecfm / 8 ) r , Stat-400 Lecture 22, Apr 12nd; 2011 Page 6 5. This is farmer Bates’s first year in the pumpkin busi— ness, and his pumpkins have a range of sizes, with diameters distributed normally with mean 131 and standard deviation Farmer Boggis has been in the business for three years: his; _ ve di— ameters that are N(14, in inches. ' 'L/lwlﬁpé/Mlﬁ‘r/C * (a) What is the fraction of Bates's pumpkins t at are above 15” in diameter? alga/,1 5W yf: [gotta/g l>(><>if> X ” NM” 22/ WW; * (b) A pumpkin less than d inches in diameter is not saleable‘ This year 5% of Bates’s pumpkins were not saleable. What is the value of d? “‘1 P< Mel) : 00f I X43 0&3) "3s 2 Stat-400 Lecture 22,_Ap1“ 12nd; 201.1 Page 7 (c) Let X be the size of a pumpkin chosen at random from Bates' lot, and let MAR a pumpkin chosen at random from Boggis's lot. What is the probability , I that Bates’s umpkin is larger than Bo gis's? x~Mi %, 22), ym/v’i / k w r/omepwgm Xr/m/VGA 2o) P<><>/) awn/>0) Stat400 Lecture 22‘. Apr 12nd? 2011 Page 8 6. Let X1, ’2, ’3, X4 be a random sample of size n 2 4 from the distribution with probability density function fxiivi : “Ema 55 E (a) Find P{maX(X1,)g;i—)‘(3) > 1) :1: [Km P(nia>('(Xz/Xz/<32 S} > Stat-400 Lecture 22, Apr 12nd} 2011 Page 9 43’ (b) A sample range is defined as R : meme-21727374 Xi— miniZ127374 Xi, find the expected value of R, Le. {H E[R]. . I ‘ :: my Xz/Xl/X 2 mMHIMMXc < 2— ] K.) W XC :m (Mme x3, 2%) LII/Z/ka , Em :— 3 ~ Hm, 2a) ...
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