Final Exam - Practice

Transcription

1 STAT 305 ntroduction to Statistical nference Winter 2004 Final Exam Practice Time allowed: 150 minutes. Authorized material :. One lettersize cheat sheet (2sided).. One scientific calculator without wireless communication feature. nstructions:. The exam has 8 pages including this one.. Answer all 7 questions; the total number of points is This exam is worth 35% of the term. You must pass this exam to pass the course.. Write legibly; give complete solutions. Marks may be taken away for unclear solutions.. You can use the back side of the sheets as drafts. f you use it for writing answers, indicate it clearly.. Please use the tables at the end of the textbook if needed. f this were the true exam, tables would be provided. Last Name: First Name: Student Number: Signature:

2 Question 1 [10 pts] The Laplace distribution has the following density function, f(x).x = ealx81 2 which depends on the parameters.x > 0 and e. The expectation and variance of that distribution are as follow: 2 E(X) = e, var(x) =.x2. Find estimators of.x and e using the method of moments. Z >..,%. ::. L 'X 2 ~ ~

3 Question 2 A Bayesian analysis is performed on a sample of n observations from a Poisson distribution with parameter.a. The prior knowledge on.ais modeled by a X2 distribution with k degrees of freedom. a) [10 pts] Find the posterior distribution of.a. b) [5 pts] Give a 100(1 a)% credible interval for.a. Express your answer in terms of quantiles you could find in one of the tables at the end of the book. [Hint: Consider the distribution of (2n + 1).A.) '1'\... X", N fo\s~~ (~} '\T"l>}'" ~~ 0: r (k/2.., Y'2) is 4k ~"\.. di.s.k~~ ~(\..~. rk f~~\"\. d.\.s.t~\ov..,\\~ is, +'h.l ~c!o..td \\ d.is.t~b(.\..t\~ of.a' f Lx,..., XV\ \\). r l >.). A 'Yt e ~2 A ~ A i."i, ~k/l e (,.. J,.)~ '" r ( i.xc ~ y...) 'H,.) ~o...'m.w\.j}... ~~t~u~ W~tl t:j!jjlo.~""taa ( ~ X:~ 4 \s\ <>Md (Vl4! \ H _ 7,) ~ 2.) S\ V\.U,NL ~~~~t\. ~ ~~O\\. is Q..~\J\ Of\..of ).... h) ~~\s ~~~ #'oj ( li~~.! ) rl (~\'\*\) A "J Z~t + k.. ~\ l \ r.y'2..", l ) 2. (z~.\)) 'Z.) 2... '"V (VZ~~Tk.. ~...' Wi lm.la ~ ~ V\AW.u. ti~. +..J ; wcu. ~ovtd.;v\ MT.1., G2,b. T~ \oe> Ld.')/b w.a.'bll ('VtftuJo.L,:s 8"~ bj tk ~.h~ i ~d f~ of ~ ~O~Wu01L. ~~~~kcm. of ~'. rk ~.f,'&..v CDw be.. ~w{ b~ t.,..a\y\o. ~ ~t ojc:.~,blv\u. ~ ~1.. ~ = f(a ~ ')\L) ~ r( ~2.~"'\)~~ tt"'~,)~.,). (:::;~.AL == ~12k~~"~' ",,x1.u)<'~" 2. 2",+' ' S\~~. w,l ~KV\.~ A~ ~ :.l. ~~t \~:. fl~~ [ X ~\z.t)(~~\c...,:.'l,\,l!)(.,y",l 2 Y'\...,. \ Z.~'" \ j )

4 Question 3 Your group of friends and yourself (total of 30 people) think a coin may be biased and gives head more often than tail. You would like to find statistical evidence to support that hypothesis. To test whether the coin is fair, each of you will flip the coin until a first tail appears and record the number of throws needed. a) [10 pts] Write down the distribution of your measurements and the hypothesis you want to test. Find the shape of the uniformly most powerful test for these hypothesis. [Hint: H should be what you want to prove.j b) [5 pts] Find the approximate critical value that gives the test above a level of 100(1 a)%? c) [5 pts] f~a = 5%, what would be the approximate probability to detect that the actual probability of of having tail when you flip the coin is 45%? How do you call that probability? 'f.'". 'X~o (~) {~~ p ~ Y'l.. P ~ Y' ~ ~~~ ~ro~~'s. Co~r~~ tc \\'M.\\~ ~ \Mo\Q.C)~"' ~ +~L".. f WCl (D.M.. f\ij~ ~) Wl wi\\ '~\ OW\. ~io'\\:' lo..\\s ~.\~ ~~~ ~ ~\W.~ ~ro+lt~i~. ~o: p'; y~. \+,., p~?, " Y2... ~ t.j~~ 1'.(j).J\A~~M.G.. ~\(s. tal)+w (=) {.u.\, of.h\jtl c:a Tlljt.c.ts "" ~, L l YL') LlP.) L G ~ c,, 2.'X' «:7 Y2.. i~f (. ( J,F, 30 A M.c..)~\~ ~\\<h\. c.f i)li. ll. L.. i \(', " '" Lc..

5 fu li.k.j;kocd ~'o fs ~ij).fo ~. zx;. '=> 0 C\\ r., ":> f ( r:> Yz) o L, ' (Go ('J NlO/\), = \ 1 ( (:.:O) ( c 1.0) o c.\\ (,0 ) ol. =") :: 2 L :=,} c:' = ~o "" ~ tmp. ~"\. ~~~J ~ \.ut of ~ ~ Mjl.ct~ ~ cff. k~~ '> (ao+~ t.1ck. c.') +...:. 0, QS:, +~ ~ tuf "ti tc:k ~ \~ ~)<.~> ~o ~ "lolls :: '12. ~'t. \... R~QL+ \\0 \ f= Q.4S"'). ~ 1"( ix" > 72..~. f 0 O.~S:),\~ l~ ~~~\.. ~o. 2. L 1) X...;;>0. c::..\ 0.JS' \~, '> :!.o

6 Question 4 n a casino, you observe a table where the game involves tossing a die. You suspect that the die of the dealer may be tricked. You record the result of each game. For game i, ni is the number of times the die was tossed and Xi the number of times a 6 appeared. You recorded N games in all. a) [5 pts] What is the distribution of Xi? b) [10 pts] Under the model above, the MLE of p = P(get a 6) is p = 2:f::l Xii 2:f::l ni. Find the generalized likelihood ratio test of level a = 5% for Ho : p = 1/6 against the alternative H : p =1= 1/6. L = \''J N. b) ~\~ ~ "'J~~\s \ MoQl~ ~o; ~: V\c, \\,', f ~ 1;(0. ~G\<\" ~t.t~ ~, 01.~ f( \{ite.+ ~o l ~o +W.,) ~ rea "~ f (h, t~ ), ~ (Y\a'). $~ L.,(p) r fj:j'lj ~ c,. ~ f (.;2lg3.A ";>2\~ c.. \v ~) 2 \~ c. ~ ~1. :, st\ ~ d.,', ~~, N A::: 1\ ( n~ \ (/i \~L (~ \~ ~l) ~(0 ) 0 ) V\i '')<.~ t. Gn r)(' (( p) "''".. N (~Y'" ( 5"/,: \""',.,<:) P lp )

7 Question 5 Consider the following discrete joint distribution x=o 1 2 y=o ' a) [10 pts] Calculate the correlation between X and Y. Note that var(x) = 5/9 and var(y) = 73/162 (you do not need to compute them). b) [5 pts] What is the conditional distribution of X given that Y = 2? Give its mass function. E ex)::' O. t" 2.. \ ;. ~ 3 to ~. El'{)~ ~ 'J l1:o\os.') \ \. \, 1. t l ' \0 'e,..l = 1.\ '1. c.cn.. (x., ~') :. ~ lx,'{) ~ ~l\l\ \J)1lNY ~,, ~ ~. 1~~2. z.fi. ~~.~ 1'( ~)(>"x, '( :.2 ') fo 'L\ V\StO.M.u.fl,~ 1'/:. 0. o :S/5" fn::2). z.... o f'c'k=e('bl \: 1/ 3 /\2.. ~ : ~ S"/b~

8 Question 6 Answer the following two questions by a short paragraph. a) [5 pts] Describe one possible use of bootstrap. b) [5 pts] Explain why randomization is important in a controlled trial. Use an example to illustrate your explanation. t)~o\~*'4' CON\..~ UARd 6 clit.jj\.'ma.v\f, +k V~(.4., of O\i\. JAt ~\\.. "E,oot 5>~ CoM. ~ w.u\ ~wc ~O~!).\.b.. RQM..Ao\\M:Co.t\Ov\.,~ r~f\.tg.mt (""" Q. COV\..tlollid tlt\.o..q +0 ~ ~!'\.OLt~ Ml. ~~~. f ~ ~f\o.6j Q1\.l \'\ot ~~~~, ~ 4 MAL,~ (~~ of 0 fmrj~t co~ b. <iu.o..~ 4h c~eloof ~ jl1o~ ~~ +~ ~ JL~t * ~ U~ (r~. VOl\.40..+, CA~l \Mew.~ Q ~ em. (),.. pill ~fl. ~ tw~. ik ~ vj"'t. ~t,i~. ~ of ~"" kj.u~ ~,,;1\ W)1\t. ~t.m.., U ~! ~C)t ~i~ it WW,. ~ AJlc..k. ~w..,, \k ~,ll W ~ 3''''''''t., ~ ClU.t.J ( 'Vhic.k "" t. ~.Q.;~ do(.$ \Aot ~ h> ~ AtA. Aick.'). 0M..6 Rt, ~~ V\A UJ O\.t.. laa.ld CAAJ 42. &. ~ tjw.t ~~... ~ ~tkd ~~'J\Q+ ~ ~o~~tid.,6~ pdl d'~i.fl~~~ J ~ wl ~""~o1 /U.L ~t fjl~ ~ c{.qs~",,).

9 Question 7 You studied the lifetime of electronic components, but since your study can not last forever, some of the components were still in working order when you had to stop your data collection. Among the n components you studied, m failed at time Xi, i = 1,..., m (in hours) and the n m others did last for the hours that the study lasted, so Xm+l'...' Xn are all greater than 10000, but could not be observed. The exponential distribution is a good model for the lifetime of these components. You would like to estimate the parameter A of that exponential. a) [5 pts] Find the survival function of of X, S(x) = P(X > x), where X follows an exponential with mean 1/ A. b) [10 pts] Find the maximum likelihood estimate of A from the sample above. Because you did not observe the exact value of each variable, the likelihood is L(>')= {n f(y,) } Lit S(Y,) } where Yi = min(xi, 10000)are the values you actually observed. e XX. rk.q.a ~A, l~ st~h~ cl~ ~tvu.vd: tho",," v..) ~ ""Q, 0J\.t. ~ ~ /W.) bvt *~ \~ \\t.,~.'o.~).'o ~ ~\'\.U.: flv'd fu v~ <,of ).. M 1f\NN{.~~~ ~\\ A e >..), "" 1 ':. AW\ e }.~~:.,... t (\ ::. W\ l.x 'l \.::., t' l>\:,/ l:"" j.t \\ (A):: >..' L () ; t fs. a... ilt 'M. '" k) /\...,... M.. 1M.. l. i = 0 fff x,':;. "' "".A,,( Z 'Ji,,=,