# mt 2 - w (ESEAZVE COFY ~ NAME: . . . . . . . . . . . . . ....

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Unformatted text preview: w (ESEAZVE COFY ~ NAME: . . . . . . . . . . . . . . . . . . .. Date: November 26, 2006 Time: 13:00—14:50 Instructor: Dilek Gﬁvenc ~.,/ MATH 230 MIDTERM AM 11 IMPORTANT 1 Check that there are 5 question and a age of formulas in our booklet. 2 Do NOT use your mobile pho e s a alculator. Turn it o f during the exam. 3 Show all your work. Correct resul planation and correct notation might 4 not get full credit._ Write your name on each s ge. c o u o c o o . n o o o . u u c a u c o . . u o u u a o n u a u n n n o o u o o o o o n c o u o a a o u o u . - u u u o n n n o . . . n c . . o o a . . . . u . o o . a a - o n - o n a - u c a .- 1. VLSI chips, essential to the r nning of a computer system, fail in accordance with a Poisson distribution ' ith a mean of 1 chip in about 5 weeks. If there are two spare chips on hand and if a new supply will arrive in 8 weeks. a) What is the probability that during the next 8 weeks the system will be down for a week or more, due to lack of chips? (/5 ,0 oz'm‘J ) b) What is the probability that the occurrence of ﬁrst failure will take more than 8 weeks? [5 ,0 9/"”’L5 ) £014, a? ck]? /ouLLv/As é lag pun/L 20% 7’ ﬂaw/u I‘n (0,14) x/V/stm (M) ' WIW Am ﬁlmy/i r ox WuJL 0/ mark 134 £7 luau/45> Q) p(5‘jS/'°V" W.W Lo. don't/é" as: ILAAN ax wan/L liﬂ 8 WM!) :2 /—-—- //555/°M be" P. ,— /, ﬂ(7§>7—) 73" 74W 0/ ?’°1 /°“ ’1 -/(/v Olsxon(,\1£:3E) / (xso)— Fora} —/9(;(=2) .21 “Z "" ’2; Z I; '5” 2;. 5_ , [ﬁleﬁa/ééj NAME: . . . . . . . . . . . . . . . . . . . . . . . . .. 2. The cumulative distribution function of a continuous random variable is given by x 2 0 a) Find the density ofX. (g pm 918 ) b) Find moment generating function of X. { 7' lac/14A ) F(x) = 1 — (1 + x)?"- c) Use moment generating function to ﬁnd E(X). ( 5 fa I ‘rr ls ) a) 35:95.), =/cx) = -€x+ e’x/Hx) d/( _,X /(x): X6 x>0 NAME: . . . . . . . . . . . . . . . . . . . . . . . . .. 3. Response times at an on—line terminal have approximately a Gamma density with a mean of 4 seconds and a variance of 8. If it has not responded for 3 seconds What is the probability that it will respond before the 5—th second? 1‘ = “=4 ?ﬂ=§ ﬁ=2 / -*é: F/2)2_7’ _.X. X' 7.. = ——e X>0 (m % '34X45> 1p (5 X>3 = P( (X/' / > P(X>3) 5 x /_Z(;. “10/ _ 3 6 W b 76 -A— \$16 Lei/<1 X 3 X -5 ,i [Le—2A; 3:— (‘Xe—Xiazéi): —Z<ia Zﬂa =- f 5 ‘% “’2 3 ,3. 6'35 ”"6. ya ’6 2+ , 2 + f(X45 “(>33 " my 7, #3— ,_____._.___. 0 +3 -_s:_’ d _' 1 ” Ejlﬁiﬂg‘ "E@{:0425 — ’3. 5 6 “’{%—H) azanﬁﬁﬁ g NAME: ................ .. 4. An analog signal received at a detector (measured in microvolts) is a normally distributed random variable with mean u and standard deviation 6, at a ﬁxed point in time. Probability that signal is larger than 240 microvolts is 0.0062, and probability that the signal is larger than 210 is 0.2643. Find the probability that a signal is larger than 250 microvolts. X / MZaJV/W/L7L algal/f 5294496 A/ m) F<X>240> : (9,0062 {0(X>2/0): 0,2643 F(Z>24o—H >: (7,0062 r aﬂﬂdz ﬂtz>2o>=/~ F530); _. ’5_ :3 F520 2/- 0,0062 : 0/9938 ’9 20‘ 2 :p M:Z,S. Q as. /&(2> w):0,2643 r p( Z>2,); /_ F(Z,)= 0'26§3 = ,63‘ 5.) F72»: /.. 0,2643 20,735? ;> Z, 0 f 2 2 oﬂﬂ= (2330‘ :) goz/Agmr /r0m Q R O 230 #64 : (aggjﬁ p: /6-0¢ N/é. ﬂ: 2¢g, (2,5)06) 2200 >2¥Z2é£9> ﬂ(x>z60>=P(2 no .; ﬂ(2> 3/3} _ /. F315} ., /M (1933/ = (9,0003 . 9 NAME: . . . . . . . . . . . . . . . . . . .. 5. Let Vand Wbe independent random variables and each is uniformly distributed on (0,1). Find the probability that roots of the equation x2 + 2Vx + W: o are real numbers. VQL.m/;:o \Jigw dM/Dz/ wévél /(W>:l o4w41 L d(v,w):i WW); WW v29” .Lvl :: c/Wo/V 0 0L 3 L ﬁ / vZo/VT-é‘” "1 0 0 £85,513." ﬁx”) ...
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## This note was uploaded on 02/01/2010 for the course CS MATH-230 taught by Professor Dilekgüvenç during the Fall '10 term at Bilkent University.

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mt 2 - w (ESEAZVE COFY ~ NAME: . . . . . . . . . . . . . ....

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