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exam2stat4710tr1230pm

# exam2stat4710tr1230pm - NDU-ll-EE’IE’IS 15:25...

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Unformatted text preview: NDU-ll-EE’IE’IS 15:25 F'.|ZIl/|Zl‘3 University of Missouri Department of Statistics Introduction to Mathematical Statistics: 47 10/7710 Examination November 2005 Instructions: Time: 60min. 1. Attempt all questions. 2. Show all workings: Points will notbe Total marks: 75 awarded unless all workings are shown. 50L 0 Tics A1 5 Student Name Student number Seluence || TR 12:30—13:45 NEIU-l 1-2885 15: 25 F' . [32/89 Question 1 [151 Tho density for a continuous random variable X is given by f(x) =§xze", O“: x {on (a) Show, from ﬁrst principles, that the moment generation function is mU) = (1-0'3, M1 NEIU-l 1-2885 15: 26 F' . [33/89 (1:) Use the moment generating function in part (b) to ﬁnd E(X ) far this distribution. Question 2 [15:] A system has eight components ShUWI) in the ﬁgure below. I II III IV V NEIU-l 1-2885 15: 26 F' 84/219 (a) Find the reliabili of the arallel s stems in I: (b) (c) (0) Suppose that assembly [I is replaced by two identical components in parallel, each with reliability 0.9. What is the reliability of the new assemb] II? (cl) What is the new system reliability after making the changes suggested in art (0)? NEIU-l 1-2885 15: 26 F' . [ES/[3‘3 Question 3 [15] Let 2 be a standau‘d normal random variable and let Y = Z Z — 1. Find thf: dansity f,, for Y. NB: The standard normal density is given by fzu) = J;— E-éz“ 5 ze ER .777 1'4ng Til; 3.4.3.1”: +545 ﬁffﬁl’ic* 75 11¢ Gd'K-JWM: I FW r I jlﬂkjl ': NDU-ll-EE’IE’IS 15:26 P [ZS/E19 Question 4 [30] Suppose that X and Y are jointly distributed random. variables with joint density function 1 fm,(x.y)=w, Oexe y-czl J’ (a) Sketch the su ort of ﬁns oint dense (b) Show that the marginal density for Y is the uniform distribution on (0,1) NEIU-l 1-2885 15: 26 F' . [ET/[219 (c) Show that the conditional clansity for X given Y = y is giwm by fx|y(x)=_5 0{x{3’ (d) Derive the regression function of X givan Y = y, La. E(X \ y). I ﬂx/j = ECXIj) if :: {nfxlﬁgég 5-5;»;- : r yaw]: J. E j J may 5.1, NEIU-l 1-2885 15: 2'? (c) Find F'(X S 0.25,?! 15 0.5) END NDU-ll-EEES 15:2? P.E9/E9 Find P(X :5 O.25,Y 5 0.5) END TDTHL P.89 ...
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