exam2_sol

exam2_sol - Solublt'ohs ‘_________.,.-..—-—l EE302...

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Unformatted text preview: Solublt'ohs ‘_________.,.-..—-—l EE302 Midterm #2 MWF 11:30-12:20, Prof. Gelfand Instructions: 0 There are 10 true—false problems (5 pts each) and 2 work-out problems (25 pts each). Do all problems a You must Show work to receive any credit on work-out problems. it Calculators but not laptops are allowed a 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: I Binomial: 101.: (k) = (1 — p)”_k , k = 0,. ..,n, where p is average arrival rate. K npflri xnpa —p). Also 2 may, 0! = 1. o Geometric: pT(t)=p(l—p)t"1,t= 1,2,...,.T:%,a§.=%§9. H I Negative Binomial order k: 191' (t) 2-— 1:11)p"(l -— p)t"k ,t m k, k+1, . . .,. T = 15;, 0‘31 oUniform:fx(a:)=b+,a<m<b.)—( 9%{g}: 12 a -- 2 0 Gaussian: 15(03): “£73m (“i ) «— 2 —- 2 __ 1 $-—.X 3-4? I Jomtly Gaussmn: ny (:r,y) = ———1—— [ 2(1‘133m) ( “X "X 2 a ‘/1— 2 — w 1r xay 9x1! _a-‘;J;.: a; _ X (y _ Y ) . . . _ 2 .._. n Conditionally Gausman: fxly (mfg) = Warp(—% ) where mXIY = X + XIY 2%1(9—?),U§r|y 2 “g: (1 *Pgw) . Leibnitz rule: g fig} f (m, t)dn: : f (b (t) ,t) age — f (a (t) ,t) Mg! + fig} Pig-91m o (I) function: 11> = 500 742-; exp (— 5552-) 032: (D x-EL'v'labn'd'MYexnobmcva-tuMn:-Inuuv-IXVTK—vl-nvlfll\dw|mflv~‘-?>NM'-:vv-w-—-—-K—H--w—- «gram-WWmmwwmmmnmmmrm 0!. Lu U.sz , Hum TthY mun {L‘Y wderwdtu. ‘ © 90 iEI-u: :l’um mama» -. SM§X[%]$‘QYCE'MM “9- it’— l’rt‘fwlllolh-pl't “Viewed? icy) rats : PrCNWWPN + PrCStNSXlP : s)€“l°l*§ig~ll’ 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). T 1. Let T be the time of the second arrival for a Poisson process with average arrival rate Then Pr (T s t) = f; Ages—“ale for t 2 0. i T 2. Let T be the time of the second arrival for a Bernouilli process with average arrival rate 11;. Then Pr(T 5 t) 2 ZE=1(t-~ 1)p3 (l —p)t“"" for t 2 1,2,. F 3. Let X be a continuous random variable, and let a be a constant. Then E [g (X) {X > a] = f5” 9 (w) fx (m) (193- i F 4. Let M1,. . .Mn be mutually exclusive and collectively exhaustive events. Then E[X] 2 23:1 E [XlMar] Pr (Ma) and Var [X] = 3:1 V3I(X1M}PI(M5) - T 5. Let X and Y be jointly continuous random variables. Then I IA ny (:c, y) dardy = 0 if the area of A is zero. T 6. Let X and Y be jointly continuous random variables with fxy (11.33;) ¢ 0 for m2 + $12 5 1, and zero elsewhere. Then X and Y can never be independent F 7. Let X and Y be dependent (i.e. not independent) random variables. Then Cov [X, Y] 56 0. F 8. Let X and Y be jointly continuous random variables, and U = X + Y and V = X _ K Then my (a v) : fxy (% (U + V) , %(U — V)). T 9. Let X and Y be independent continuous random variables and Z = aX + Y where a is a nonzero constant. Then fz 2 If; fx fy (z — x) a 3 F 10. Let Y = s + N when a signal is present and Y = N when a signal is absent, where 3 is a i positive number and N is a Gaussian random variable with mean 0 and variance o2 (and N j is independent of Whether the signal is present or absent). Assume the signal is present with probability p. The detector decides the signal is present if Y > 7 and the signal is absent if l l 3 Y S 7. Then the probability of error is (I) (1 —— p) + ‘15 3). News: 1, T 1'9 3.“! crib“ RAM?) 1. T 1'9 1“,“ uvflltr Megan's; Ecmamffiifilso PrlTétl: f (qursupi's ; ‘5 ltflll’gll'hl 3:7. 8:! U0 ‘5- tfgmmflz J: am lxlv‘ X>MM = 5 9M lfiyfimlffi‘ 0L '5 lac ly’lllx‘rl do: “*3 WWW) " i “Pvt.er ' h I . Var—DI): a Vuvtx‘Mij Prch-)+ a @[Xiht‘lrf£xj) lrlHLl ' ' z Q. t:: 1.3! C. Leann ' 5. i’XN (“Infill O‘Hahubd )g. (udlptl/Lalwbt . :- tuutova-M‘d) 2 I l '3’- C""“'” o I .L' in» Jt l’” "31“ Sf ‘ 7‘: ECM+V))1}:,LUI'Vl W 1“) “ 1-1 '3 Questions 11 and 12 are work-out problems (25 pts each). You must Show work to receive credit. 11. The pair (X, Y) is uniformly distributed in a wedge centered at the origeu with radius 7'0 and angle between 0 and 60 radians (measured relative to the cc axis). Let R and (-3 be the radius and angle corresponding to (X , Y) . A “J— ‘Aa [- Q) Are R and 6 jointly continuous? You must fully justify your answer Cb) Find fag (r, 9) . c3) Find ffior) and few). (at) Find E139] . 05.) Find fR(r|R > get). : -‘:7:._.7-..._.‘..... J“.!.. |.-- unguarv'gvnvxw'v ,. - E Comfihweug) at 3ofmrb} tom-Imqu 093’). 9(th GR, {9 $9M:on iii fiwugh'ew C e : 8? (m L Hi9 r' ) M U“ mm 9‘5 i“ H“ how-amourh) a, (g [M‘fu i—o Brand Jtr‘e) 3"“4‘59: “L q Nd“ qt: route) «3.: rsan So apt—rig) use -rSa'nS‘ \ YCG§ 3%,?) u 035 ‘63 -= c 3r ‘3? SB. 3’3 b} S‘iHB rm 3; ’69 t r <‘ < Heathen—9‘” ) 03'1"“ “9‘?” / Ft‘ha“ QwiuN-e ConQ+W c— g Q we 9.! _r~o 0 our _ Cr 30 r T- C, V‘ (19 ‘3 SS PRI®CFO)d-90\ So So 1 6 0 Q s? 0: 3;: // ¥ Mrs 96 89° 2. r dig ’- E: U") :- d9 :- H _ 7' R yo ER, @(r'm 0 flag r0 :Sl@te')hr 5'90 (9 Che) ir -: 5Y0 7:; r the = l— o ' o rev—9° 90 i a“) gem (Cb-t4 alga use out +0 g” HA“ GFHMV' Stmcl (rte) e K wdwws‘r wane): mm We») . arr EEK®1tkg§R Erma“? = iniflfle ii, rdé’a‘dw I 7. Q” G a? 7- “ Go ‘5 o o r‘ g t v; i 5 'r a :11" 1’. ~-—-- r3 gap 0 o r" 9" 3° 2.6 “at 3 1 "ti—oi — r:wt.Hmummmnwmmm.wmnmmmm‘muflw .. ( (Um MW M“ R,@ mammwdem Jr0 mutual-e 9.5 Em HM]: EUGENE) (é) ¥RCHR>K§).¢ Lam r>Vo 3 H s Pytk‘r‘rg y 1_ 1 E ‘3 r 00 7.) Va 1‘. dr d git 6.; he :9 5 rtR>scyer=§ — -1\., a g a i z 3. Y E ¥ (r1 H F?- } “'x \ R z 3 r0 2, i i a i ;. a»:invumw.mm?wam. 12. Let X and Y be jointly Gaussian zero—mean random variables with variances 0:2," and 0%,, 1 and correlation coefieient pr (lpxyl < 1) . Let U = aX «1- 51’, where a and b are constants (b 79 0) . You must express your answers in terms of 0&5?“ pr, a. and b. (a) Find 17 and 0%; . (b) Find pXU. (c) Under What conditions are X and U uncorrelated? independent? 3 ((1) Find and sketch fU . (e) Find and sketch fXIU 1 (f) Find the MAP estimate :1? of X based on U = u. Efaxib‘flc 00?qu = b// Mgr ewe—mom) w c :6: Von/[mum]: misfit bzvfar am) Cou Hm) pm Jaw: mm” a? Cw her]: ymnoflf “X W 3 . (Th1; wank-£6 1+ embny sxoflffl m 3w ~.- 016“, (THE. : QO‘XL-l- \anxYGx 5? _ / TX -\v blfi'YL is WY / CM MY 3mm»? awn r> um 305MB} Gin/«A99 (L‘mww Cflmba‘hah‘wv‘jé s m w 56*; (it): U '*’)("‘> umzm Y I 0 .- 90 u I )6 “Va 359:“ HID cthL‘MM “g Aflt‘MI-Lq) > . 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exam2_sol - Solublt'ohs ‘_________.,.-..—-—l EE302...

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