Final 2007 solutions

Final 2007 solutions - STAT 116 STANFORD UNIVERSITY...

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Unformatted text preview: STAT 116 STANFORD UNIVERSITY Department of Statistics FINAL EXAMINATION STATISTICS 116 Instructor: Professor Anthony D’Aristotile August 17, 2007 3:30 pm - 6:30 pm IMPORTANT: Please, show all your work and justify your assertions. You may leave your answers in indicated form. There are 200 total points. STUDENT Name: STUDENT ID : INDICATE IF SCPD STUDENT (YES OR NO): 1. (10 + 10 points) Suppose that the heights of male Purdue students are normally distributed with mean 70.5 inches and standard deviation 3 inches. (a) What is the probability that typical Purdue male student is taller than 75 inches? (b) If you randomly sample Purdue male students one—by-one until you get one taller than 75 inches, what is the expected number of men you’ll need to sample? ?<2Z> 7S'-37o-§'> _: V(2> ___ ?(2,(.Y) P< X>7$> : \_ 7(2615); 2. (10 + 10) (a) Suppose that X is a random variable whose moment generating function is given by Mx(t) = 1/5e‘ + 2/56“ + 2/5e8t for —00 < t < 00. Find the probability distribution function of X. (b) Suppose that the random variables X and Y are independent and identically distributed, and that the moment generating function of each is et2+3t for —oo < t < 00. Find the moment generating function of Z r 2X — 31" + 4. qfiz-I-‘i‘ _ “(170- ‘2 zQ" Hzx‘t) x 911-91‘ H (E: "(031%) = .9. -3‘! 4t¢+61+ §£2_91_ .2 p __ M (ae‘)‘- "- x Y»; A "-31 1112-37‘" ‘5 .11 44 (*3 3. (20) A miner is trapped in a mine containing 3 doors. The first door leads to a tunnel that will take him to safety after 3 hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of travel. The third door leads to a tunnel that will return him to the mine after 7 hours of travel. If we assume that the miner is at all times equally likely to choose any one of the doors, what is the expected length of time until he reaches safety? - 1' 4 “.dggcflmu \ M . 0 = 2 h u 2 y’ 4M 3 \\ “ 3 I} AM 3 3 _|_ 4 <g4E(X))—; '3 9 J—E(X) __4- 5+ Lead-i 3 3 = ‘+ 3 7 IS‘ 1 Z+—= -—- E00“- 3"' 3 3 3 7\ €LX)=‘S' II ll (14 + 6) Suppose that balls are withdrawn one at a time and without replacement from an urn that initially contains n balls, of which k are considered special, in such a manner that each withdrawal is equally likely to be any of the balls that remain in the urn at the time. For 1 S i S n, let Xi = 1 if the ith ball is special and let it be 0 otherwise. (a) Prove that the variables X 1, X 2, ..., X” are exchangeable. (b) Find the probability that the first ball is special given that the last ball is not special. Explain your answer. ‘ I“ t.’--—’~h cg“ GM‘ P(x|=?l) x1: ‘) -”’ Y’=1h) :- ft5=k a any. x ,-. 3‘ MK 1% (mm m M4 —~ w M Md“a"'m* 5. (10 + 10) GE has 2 plants which make 75 watt bulbs. Their plant in Churubusco, Indiana, makes 60 percent of their 75 watt bulbs and Churubusco bulbs have exponential lifetimes with mean 3 years. The remaining 40 percent of their 75 watt bulbs come from Tuscaloosa, Alabama, and Tuscaloosa bulbs have exponential lifetimes with mean 2 years. (a) Find the probability that a random GE 75 watt bulb is still burning after 5 years. (b) Given that a random GE 75 watt bulb is still burning after 5 years, what is the proba- bility that it is a Churubusco bulb? C —W' Lewdgce 95 “aw T__7M.Jnut 5 07" P( c)‘ 9(Lvrf“)° “L; > L> ' tad-V) )?(L>€\<3 + HT)“ —_ MC - i l’. _4 J‘ i .‘(A 3 ,3/3 -: (.3 Ml . 4 = ,g; “3+: 7:“ 6. (8 + 8 + 4) Suppose X and Y have joint probability density function f(x,y) = 6y when :1: > 0, y > 0, and a: + y < 1. (a) Find P(X > (b) Find the marginal probability density functions of X and Y. (0) Are X and Y independent? 7. (20) Let X and Y be independent exponential distributions with means E(X) = E(Y) = 1. Find P(Y22X+1). N ‘12 a»): S 4*” [inffia #31 PM o [ff/‘1 \\ _ly-‘ II Vj e \o I 7: l g’ I, b I m ’ c U I u if 8. (20) Suppose that a random sample X 1, X 2, ..., X 300 of size 300 is taken from the uniform distribution on [0, 12]. If 5300 = 3101 X j, use the Central Limit Theorem to estimate Pa 5300 — 1800 |g 15). 1 _ (2+) - to \{i E(><)=é “"7 it X ,,.{o~v~ F” ) - i3; _ n 3.. fl Mk goo.é : lQoe - X‘ “L c {380 - Z‘ 1 m4“ goo.\z '—‘ 3‘ U u )4 A” M (:11???) u ’ < \Sm- ‘9°°‘ f _ 1’ ~41” _ wéoif is) _ Y< 15w- f.) A v < ‘2” 4 43') .g 2’ O z ’ 2 H - H74 2 (fig?) O 9. (10 + 10) A seven judge panel will make a recommendation in favor of or against the death penalty. Each of the members of the panel will independently vote in favor of the death penalty with probability .7 and against the death penalty with probability .3. (a) If the panel’s decision requires 4 or more votes of support, what is the probability that the panel decides in favor of the death penalty? (b) Given that exactly 4 of the judges voted the same way in this matter, what is the prob- ability that the panel decides in favor of the death penalty? H J10 (a 0 VA ’2 w V“. A \J h 3/ -h D \‘D W 10 { (mu) -. '7;— (N'fi) Q1”) ‘5 ’T “(3) (“)V) 0 ch»!- 4 10. (4 + 16) Consider X and Y with joint probability density function f(x,y) as given in Problem 6 (a) Give a geometric description of the domain of f(x,y). (b) Find the joint probability density function of U = 2X and V = 2X +2Y. (HINT: In the any-plane, consider the triangle with vertices (0, 0) , ( 1, O) and (O, 1) and its boundary B. _ —1 —1. “21433402233 In the uv plane, find T (B) where T - v = “x, y) = 2n: + 2y maps the cry-plane into the uv-plane. ...
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Final 2007 solutions - STAT 116 STANFORD UNIVERSITY...

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