2007 fall mt1

# 2007 fall mt1 - Hﬂfﬂ 235\L «AME Date Time 18:00-19:40...

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Unformatted text preview: Hﬂfﬂ 235' \ \L \\ «AME: . . . . . . . . . . . . . . . . . .. Date: October 19, 2007 Time: 18:00-19:40 Instructor: Dilek Gﬁvenc IMPORTANT 1 Check that there are questio ' r. Turn it off during the exam. . sufﬁcient explanation and correct notation might GOOD LUCK! u n n n n o c o o n c o o e . o o c o o u o o u o c n o n o n - o n u g u . n n u a c a o o o c o a a o y o . a u . . . u o a . - n g . o n o o o n o c o u o a o a n u e o a o o c u o . o .u 1. Suppose that a random variable X takes values of O, l and 2. For some constant c, P( X: i ) = cP( X: i—l) i=1 ,2 . Find the expected value ofX in terms of c. fow): CVCXEO) _ ‘ r FOGZ); c For: I): c [ C ﬂCX'soﬂ ; (12/4090) ﬂ(x=o>+/(le)+ W292) : 1 /(Xso)+ exam: 0) +cZ/’(X=0) = i (3—?) NAME: ........................ .. 2. Consider a time division multiple access (TDMA) Wireless system, where the base transceiver system of each cell has n repeaters. Each base repeater provides m channels, thus there are mn channels in the system. A base repeater subject to failure. In order to evaluate the impact of such a failure on the perforrnability of the system one should know the number of ungoing talking channels on the failed base repeater. Suppose channels are allocated to the users randomly. If there are k talking channels in the whole system, ﬁnd the probability that z' ( i S min(m, k) ) talking channels reside in the failed base repeater. RESERVEA NAME: ........................ .. 3. a) If A and B are independent events, show that A' and B'are independent events. (A' and B' are complement of A and B, respectively) (10 points) b) Suppose a balanced coin is tossed two times. Deﬁne the following events: A ={ Head appears on the ﬁrst toss} B ={ Head appears on the second toss} C ={ Both tosses yield the same outcome} AreA, B and C independent? (10p0ints) a) A owl 6 are llqaéfmaén7L=3 ﬂrﬁﬂg):p//4)/9(£) W6!“ 3/123“) M pm/ﬂgx): ppm/3y: /_ ﬂ/Aug) = /_ ﬂm)- Fwy/04’”) : PM’)— pc£>+W/W’/5) 3 pm’) , PCEMA PM” 2 pm’) _P/3)/’07’) 3 pm’)[/' PUSH = ﬂ[f7’).ﬂa?’) a) S=§/¥#,/I‘7) 7% TT} NAME: . . . . . . . . . . . . . . . . .. . Suppose that when in ﬂight, airplane engines operate with probability p independently from engine to engine. An airplane will be able to make a successful ﬂight if at least 50 percent of its engines operate. For what values of p four—engine plane preferable to a two-engine plane? (LWH 4-03am %M V wazcess/u/ /f)/3[/—/’)+/f)/94 MM 206%; Fla/be. F (Mayer MM) = (mom/a) + [3,292 5P2//'F)2+4/3(7-/37‘/4) 2/07”) 7792 Mo 6/0 [MW/Z) + éﬂz— 4/3+ﬂ3> Z-Z/D—f/a 4042/93 6f3+ 4/3712 w" > O 3/» 872+ 7/— 1> o (ﬂ~—/)[3/42.. Sig-#23) 0 09-1) (3/..2) 00—!) = (p451 (3/'*2> >0 \$4M» 60-0730 7:) 3/-2>o =3 /a>_:.— RESERVK; NAME: . . . . . . . . . . . . . . . . . . .. 5. Assume that the probability of error—free transmission of a message over a communication channel is 0.8. If a message is not transmitted correctly, 21 retransmission is initiated. This procedure is repeated until a correct transmission occurs. Such a channel is often called a “feedback channel”. Assuming that successive transmissions are independent, a) what is the probability that no retransmissions are required? (6 points) b) what is the probability that exactly two retransmissions are required? (6 points) c) what is the probability that third error-free transmission occurs before the ﬁfth transmission? (8 points) (Deﬁne the random variable, determine its probability distribution and solve.) lxxaay Awmwaem wwﬁd A,aw.yg/wﬂa¢‘ Véfuns’m‘ssﬂon ‘ Av) #093) = (0,2)1(0.8) 25,032 a) y,» #1» % Wsm‘kribng fefux-Poo/ 7(0/ 3—401 Jamar- fa WSWHMQA‘ y/v gm [/g3, /;c9,f) 4 3 3—3 v ~I Hug): (g; («a wow) 3 . 3 c. («9.8) +3/c9,g)(o,z) 1 09fo g) = (13/92 RESERVE; ...
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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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2007 fall mt1 - Hﬂfﬂ 235\L «AME Date Time 18:00-19:40...

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