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Unformatted text preview: NDUllEE’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 NEIUl 12885 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) = (10'3, M1 NEIUl 12885 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 NEIUl 12885 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)? NEIUl 12885 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;— Eéz“ 5 ze ER
.777 1'4ng Til; 3.4.3.1”: +545 ﬁffﬁl’ic*
75 11¢ Gd'KJWM: I FW r I jlﬂkjl ': NDUllEE’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 yczl
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) NEIUl 12885 15: 26 F' . [ET/[219 (c) Show that the conditional clansity for X given Y = y is giwm by fxy(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 55;»; : r yaw]:
J.
E j J may 5.1, NEIUl 12885 15: 2'? (c) Find F'(X S 0.25,?! 15 0.5) END NDUllEEES 15:2? P.E9/E9 Find P(X :5 O.25,Y 5 0.5) END TDTHL P.89 ...
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 Fall '05
 n/a
 Normal Distribution, Probability theory, probability density function, moment generating function, standard normal density, moment generation function

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