STAT1301 T9 solution

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

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern