Exam2a_Solutions

# 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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