132B_1_HW_solutions

132B_1_HW_solutions - EE13ZB Professor Izhak Rubin Solution...

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Unformatted text preview: EE13ZB Professor Izhak Rubin Solution for Homework No. 1 Set No. 3 Problem 1 Set 9 =1-p. Problem 1.1 E(N) = 2n.(1—q)q" (1 ‘01:"1": III. a d (1 - 9): -¢"- .2}. a. Sinus the motion ad the difl'emtisfion Ire interchangeable, we have EU?) = (1 - m; 21" II A H I In \II' "T II A 0- a 2’ plan. F. J fl ‘1 II “I Problem 1.2 mm — 1)) 2 no: - 1) -(1 - m" (1 - 911' E "(n - 131“" III n-ufiféZa‘ d“ 1 1.. 1 _ [ ale-1d! 1-; [1 1)»: (1-1? 24: 1-1)" ‘— Since Vin-{N} = HMN - 1)] + 51m - [smut we hm Val-{N} thlun 1.5 For Hats} 3 ID, 'l'l In.“ FNII} Prom-u 1.4 513.“) _2¢'_+L_._"_ {-fl' 1-! {1-11’ 1 1-1) 1-? T = 39"“) Z r" -{1 - 111“ lull (1 — fl Eta-'1‘ Since Var-(N) é E(N') - [E(N)]’, we have __ 9+9“ 9’ MN’ ‘ (Hm u-qv _ q _ (1-9)a _ 1.12.2 _ f. Problem2 ProblemZJ ELY) = [1:de(2) Using the integration by part, we heve E(X) = —ze""|3° + f“ e""dz 0 Problem 2.2 , on 50:3) = / emu) o = —:'e‘hlfi'+/«2se""dc o _ _.2_=_ -At on f. 1 -.\: .. A e lo 4- o Ae dz _ 2 — :5. Therefore, we he" mm = Jam-mm? 1 :5.- Problem 2.: in: Rep) 2 0. we hm Fx(e) E(e“x ) a f e"'Ae'“d: n A A+s 3 Prablmn 2.4 ' II .5 EIX‘] II. 13". tum if a. 'I'harefore. Valli)” = 1*. Problem 3 Marlin-d 1: P(X+Y=n) . 2P(x=n-MIY=m}-P{YIM} fill = Erzxsn-m).r(r.m) all. "‘ i-l: ”I'll c-Ar}. = 2 {II-m}! I In! a E-(hxi‘ArIEAx +1”): i n! All-lag n! all (n—m]1m|'(Ax+A~yP _ run-1r)“: +lrr i( I )l A? III 1 Jay u-nu a: .... u- may] [Hm] I g'(h+hr}§AI-+AYE‘ n1 mmmnfifihx+vhmmmfinfifiwmlx+lp Mlthod I: Iflletludnuldnmmmmdthmhflflummhm radian thIII-df.mflvnly. Inn-”(:1 mmmmhncfiadthpnhhfliwm- fumhnnfurthn-du-I'Ifilbhx-lvl’. HXnd Y arc-minded]: independm. fl1+r(t}- "(fl-"(1). hfithhmflunfifikIuflhnm ”(3} = if" * w -1“ 0' dug" n = a El! L13! ‘ “II 4 ' t-lxfl—I]. ram-n1 arr-[*1 Thnfou. ”wit! = ”(ti-mar) = l-{lx+HKI—l}‘ Ninth: uniquena- um: yum-Ilia; mm. we conclude In: the random variIth+Y bu the Poi-an dinirhutiun with pal-Imam A: + Jay. Problem 4 Lit Ndmtelhe numhcrofcmmpudq. magma. mlol'm Ipantbythlithmhnmfuti=fl.1,u-. Snip:flXflfudli=fl.1.---.Thm1thlwtflmufthe mmtbycuflmlpudul’ilmn-nd. N Y=£Xp The mind “In: a! Y is obtained u fallen: 511"] N FIEXIJ N = $12 x; I”)! N 3:23:an til N HEM #EW} annual-u}. Problem 5 Dim-m radon: MM-Xud? minW,if|fléonlyHP{X=m,Y =n]. FEE-mJ-Pfl’lnl. mmpmmwmrummmmmmmx. II It 1| Pfx=mj = iPl'X =m,Y=n) n-fl fl E'Tl'l'll". Em G-d‘m at l—lafl-fl = ml 3;, (n-m)! e—‘4II- III] I .— — hm=oull for n = 0.1. independent. - - -'. The marginal probability mass function for the random variable Y, " -74m3n-m C Em m=0 5—2?" ”(3. )ET [1"1'” ewfrm n! PU’ = n) Since P(X = m. Y = n) 1: P(X = m) - P(Y = n). the rmdom within X and ...
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