STAT1301 T9 solution

STAT1301 T9 solution - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: / THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 9 EX.1 Let X ~ Poisson(a), Y N Poissonm) be two independent Poisson random variables. Find (a) the moment generating function of X; the pmf and the moment generating function of X + Y; (c) the moment generating function of X — Y. 28' my; W feel, 2/, w mm): H8“ij 89'th = E68: W 3 6(6)“; gfewj I 56 7U 2 eflfiiufl‘xme‘i/J :flM, X147: 5) z from W4) ‘.’.—0 , 2 we“ 3%”? ~- r (‘WWe/WE) A 401%) 2 al 72753.20 t, (KLO W2 my” vb! X11 Ti [6({7‘6626 [ally/7755) 1 azef~U€Pth :8 EX.2 The inside diameter of a cylindrical tube is a random variable with a mean of 3 cm and a standard deviation of 0.02 cm, the thickness Of the tube is a random variable a with a mean of 0.3 cm and a standard deviation 0.005 cm, and the two random variables are independent. (a) Find the moment generating function of X N N (a, b); (b) Hence, Show that if X ~ N (a, b) and Y N N (c, d), where X and Y are indepen- dent, then X + Y N N(a + 0,172 + (12); (c) Find the mean and the standard deviation of the outside diameter of the tube; (d) Using the Chebyshev’s inequality, find the lower bound for the probability that Wis within 0.05 cm from the mean; (e) Evaluate the probability in part (d) by assuming that the inside diameter and the thickness of the tube are independently distributed as normal. a) xiv/um) ML d) 52 mama-WM $095 «if; 7?; /‘°6<9(<oo WC/N’ngflfi') A“ a S r 1 2' " Ll Ci: mji “i X: > 7g]; HR ‘ 5 km fMt/zéfifxy / 73 L on . “(ML :2 ,. _‘ ’ hearse/Z MW / New ~ 0—61 a or ,Jfixmimml L 2322629” ’ “t r if: rise 5 f ’2? Ci” . g L “(fl-Mar 2; 04 { ’ 262% J" gigallfl mawz 2r 1% L q u Q) DNA/(3,002.) M xix/fig lag) ‘ 7”“ w“ my ” A‘ r "N C( &y{ _ f. I ’ , : +2 mNCléfl'WS) Hare : rim/(7r) u ‘ N y ’ zeafrégfi’ ecffi‘f flaw VON/flk 0.05“) I ewwcfi‘éi’iL 1/76 lpj’éflkrfl. } ['0' XWW/VMlcjmv ' 9 2 :zécuwv/ C we we, 24% MI in " zero-87:94 7-— ie mange we 2 033% i more. MW N: 9+2? #5660 imam): 3 12(06):}.é a is Vii/(7‘) = M» (1)) 7‘4 VWKT)=&02L+4(Q-005’) = amrmb 19’: )p,my = 0, 02234 CM A EX.3 (a) Given Z ~ nb(r, p) with mgf (L) , show that if X, Y 313 geom(p), then“ 1—(1—p)et X + Y ~ nb(2,p). (b) Hence, find the conditional pmf of X , given X + Y = 71 When X and Y are in- A Xff fidflWC/J), f < :240772 3 : [z—if‘ojp) '3 XW‘MMZf) 27) V527(X:’X [X‘H/zn) POH‘fzn ) I: 1L=nj : (I’M—’1’) (X422?!) «1 :- (2‘ gmxer, WWW) 2"” U77 1%): 553/926/72’”) ” V 3 H'énca X 9647:” E5 dist/“flaw A; &A% [7/2, - » ~, fl-U, EXA Suppose the distribution of Y, conditional on X = :17, is N (:13, 2:2) and that the" marginal distribution of X is uniform(0,1). Find (21) Em; A (b) Vai‘(Y); and (c) Cov(X, Y). 1%) Em: Elm/>0} ( 1‘ Y/ )(=9(' WWW) : ECX) W 2 (726/ “126 (9} V i] VH7): I/arZEKT/XJJ 7‘ ELM/{HM} ’ , ; yaw/1L E00 6W7 i/a m, A I J, 2 '1 Bi, ” E “f 5 — f “E j V ‘7; a) éflvéégflwfl :QZXECYWJ :2: (X7 , ,L " 3 ...
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This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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STAT1301 T9 solution - THE UNIVERSITY OF HONG KONG...

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