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EE302%20prevexam1a

EE302%20prevexam1a - W(145 EE302 Midterm#1 Section 2 TR 12...

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Unformatted text preview: W (145 / EE302 Midterm #1 Section 2, TR 12 — 1:15, Prof. Gelfand Instructions: 0 There are 10 true-false problems (5 pts each) and 2 work-out problems (25 pts each). Do all problems 0 You must ShOW work to receive any credit on work—out problems. 0 Calculators but not laptops are allowed 0 Cheating will result in failure of the course. Do not Cheat! 0 Put your name on every page of the exam and turn in everything when time is called. Useful formula (the mathematical formula in Appendix A of the text is also appended to the end of the exam): 0 Binomial:I Pr (X : k)_= (71:)pl” (1 — p)n*k, k 2 0, . . .,n, where p is probability of arrival. o Exponential: fx (r) : APT)”, m 2 0. X : f /\ /\.r “1’16"” 0 Erlang order n: fx (”5) = (#1)! ,m > 0. Y 2 >43 0 Uniform: fX (1') = fi, (1 < a: < b. (27f) - (D function: <1) (:L') = ffoo L exp (_%2) dr 27r LOD-d 0 Gaussian: fX (:15) = l V27702 exp (— Questions 1 - 10 are true—false problems (5 pts each). Label each statement true or false to the left of the problem number. (note: if statement is not always true, then it is false). E1KJ\WW°-HOV‘S ‘ F 1. ‘2 2. T 3 T 4. F 5 1- 6. F 7. 1— 8 T 9. T 10. t. A — Pr . P (AURUL) — PHAH ”l r + Drmw— PrCU" PrCEnc) ' Pr((m\!‘$)u(nnc)) iMA—B): (9,-(PmTZ): prtmmrui) : Let A,B and C be sets. Then AU (Em C) = (AflB) U (Am 0’) Let A, B and C be events. Then Pr (A U B U C’) : Pr (A) + Pr (B) + Pr (C) — Pr (A H B) — Pr(AflC)—Pr(BflC). . Let A and B be independent events Then Pr (A — B) = Pr (A) (1 * Pr (B)). Let A and B be events (Pr (A) > 0). Then Pr (BlA) : Pr (AlB) Pr (B) / Pr (A) . . Let X be a random variable. Then Pr (X < .72) : FX (33) . Let X be a continuous random variable. Then Pr (X < 3:) = ffoo fX (x) dx. Let X be a discrete random variable which takes a finite number of values, and Y = g (X) Where g(:1:) is not one—to—one (i.e.7 g(ml) 5A 9(332) for some 331 9E 132). Then Y is a discrete random variable which takes fewer number of values then X. . Let X be a continuous random variable and Y : g(X), where g(:z:) is a smooth strictly increasing function (i.e., 9(321) < g(mg) for all :51 < 3:2). Then fy (y) : f); (cc) g—j when y 6 range (9) . Let X be a random variable and a and b be constants. The Var [aX + b] 2 a2 Var [X] . Let X be a random variable. Then Var (X) Z 0, and is equal to zero if and only if X is a constant. u (Mo = (Av8)n(bruc) (Bur) ' Pr Centauo) ‘ Prlfi) —“ PHHH 9MB) r 9r“) - Pr (gm) “(PM/5mm + prcp‘ne) , yrcnnfénc3) Prlxcx)’F(7~) + l’rix‘ix) Ur ?r(7K-«oc)>e PM“ WW“ it (”cud—5 MAW-SS ‘3'“) E»), )Q: end-xx” Amwgk W5» rub“; rtdz (bud ”‘3 bwhqfl CHM/2 ?h(A)(\'Pr-(§)) he 3:33 0d" MM Cl HM “‘4 Wu): K com tabt J t *- w $29.me 6%- Questions 11 and 12 are work—out problems (25 pts each). You must show work to receive credit. 11. Do either A or B. A. Leif Wrov‘ S-rw Mzsswge) u; M“ 8— FWVSWCS Computer A sends a message to computer B simultaneously over two unreliable tele— phones. Computer B can detect when errors have occured in either line. Let the prob— ability of message transmission error in line 1 and line 2 be Q1 and qg, respectively. Computer B requests retransmission until it receives an error—free message on at least one of the lines. Errors in different lines and/ or at different retransmission times are all independent. Find the probability that at least n transmissions are required. (Hint: the probability that at least 77. transmissions are required is the same as the probability that a certain event does not occur in n trials for a particular sequence of Bernoulli trials). . A multichannel microwave link is to provide telephone communications to a remote community having n subscribers, each of which is active independently with probability p. The link can provide up to k seperate channels, each of which can accomodate a single subscriber. Find the probability that the link can accomodate all of the 71 subscribers, in terms of p, k: and n. (Hint: the probability that the link can accomodate all 77. subscribers is just the probability that the link can provide a channel for each of the subset of n subscribers that are active.) — 4. )k Xi‘— l—Irowsvw-‘35f°”5 at: h A— Lwcwauvtfi S‘irs’r ‘ 3' N: it ewuw 51mg WISSD‘S‘W {tIW( ‘ ’n/W'V“ QrQK>/°"\ ’ 3.. k v ra 0.8 c 25 WW 12. Let X be a random variable and Y = g (X) , Where 2 . a: w>—1 g(m)={1’ mE—lj Suppose X is a uniform random variable between —2 and 2. (a) Find and plot the probability density function and probability distribution function of Y. (b) Find the probability that Y Z 1. (c) Find the mean of Y ((1) Find the variance of Y. .39”. it; . In *7. 1‘; 'x ”L ‘L _ 3L -.) + 9t») (‘0 91W): Vr-C‘hfl I? C \d:c\ 0,334! (g (a): ¥X(*I)lo:—:gl " gxm“) *3 *3 -2 d. Vrf‘f:;\: Pr()ge-|)_5-bo¥$(m II~ ‘ , am; ._. .4 we: we, «3-43) us" as 5M” g“ I; a,” Euro J, rem 'do ~ .L §L331C§XC-fi3+§xflfifl[email protected] omjct .. 5x (‘55) L }>\ ...
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