Exam2a_Solutions - NUU—lfil—dlfllfib llfl: dd |-' ....

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Unformatted text preview: NUU—lfil—dlfllfib llfl: dd |-' . Ell/UH University of Missouri Department of Statistics Introduction to Mathematical Statistics: 47 10/7710 Examination November 2005 Instructions: Time: 60 min. 1. Attempt all questions. 2. Show all workings: Peian will not be Total marks: 75 awarded unless all workings are shown. Student Name 5 a? L U Tr CM! 5 Studant number — Se uenee MWF 9:00-9:50 NUU—lfii—dfllfllb llfl: Jud |-' . Eld/ Elti Question 1 [15] The density for a continuous random variable X is given by fut) = fife", Dc: 1.: -:'. an (a) Show, from first principles, that the moment generation function is mm a (1%)“, 2: <1 NUU—l4—dEIEID llfl: dd |—' . EIJ/ Elti (In) Use the moment generating function in part (b) to find E( X) for this distribution. Question 2 [15} A system has eight components shown in the figure below. I II III IV V . 4/lfilti NUU—l4—dlfillflb llfl: dd l" Id (a) Find the reliabilit of the arallel s stems in I: E}: g ;_ (t-o-e Ir- e.“f)(r— as?) E y _. «:3. CD ('3 E; a s ‘7 ‘5‘ 4- (b) (c) d-T‘?#-w of? x e.5"x a. 5’6 x: m ’3’ d-éifiz (e) Suppose that assembly II is replaeed by two identical cempnnents in parallel, each with reliability 0.9. What is the reliability of the new asSeniny TI? ((1) What is the new system reliability after making the changes suggested in 311(0)? EEKEIXEEH ,JILE'E' awe; ,x a??? 7: ctr—3' >-= {33% Xe”? a. 639 G) NUU—lfil—dfllfllb llfl: Jud |-' . EI'J/ Elti Questian 3 [15] Let 2 be a standard normal random variable and let 1" = 22 +1. Find the: density fr for Y . NB: The standard normal density is givan by fz(z) = ~1—e_‘§22 , 26 ER fl mg. (75$ H14. Sch-aye Tail-9459‘ £9) + {gaff}: -, - é.(ff—’/ \lzaéja—n Cf J 33"" NUU—l-fil—dlfllflb llfl: 3:: |—' . Ub/lflti Question 4 [30] Suppose that X and Y am jointly distributed random variables with joint density functicm 1 fw(x,y)=—-, 0:); fix<2 2x (a) ' (b) NUU—l4—dlfllflb llfl: JJ l“ . m'r/ lilti (c) Show that the conditional density for Y given X = x is given by I fy;x(}’):;1 Ofiydx (d) Derive the regression function of Y given X =x, Le. EU” | x) NUU—l-fil—dlfllflb llfl: 3:: |—' . Uti/lflti (e) Find F(X g 0.25,? 5 0‘5) TEITFIL F' . [38 ...
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Exam2a_Solutions - NUU—lfil—dlfllfib llfl: dd |-' ....

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