ELE 509 Final Exam Dec 2018 1 Cnsider tw Gaussian randm sequences X[n] and Y[n] Assume that they are independent f each ther with means and autcvariances μ ' 3 μ * 4 C ' [m] 1 2 1 3 and C * [m] 3 1 10 1 9 We frm a new sequence via additin Z[n] X[n] + Y[n] a Characterize Z[n] by prviding its mean and autcvariance functins b s Z[n] a Gaussian sequence? s it statinary in any sense? c What is the pdf f Z[n]? What is the prbability that Z[n] > 9? d Unbeknwnst t yu there was a lag in the system s that the samples were nt quite lined up in time; in ther wrds the new sequence is really Z[n] X[n] + Y[n 1] Hw des this change yur answers t the abve questins? Hint be careful f the nnzer means 2 Let X[n] a WSS Gaussian randm sequence with mean μ? 0 and autcrrelatin functin C? [m] @ σ B m 0 0 therwise be the input int a simple lw pass filter with respnse Let Y[n] be the filter s utput h[n] (05 P u[n] U Y[n] R(05 S X[n k] a Find the cnditinal mean f Y[n + 1] given Y[n] E{Y[n + 1] Y[n]} b Find the cnditinal mean f Y[n + 2] given Y[n] E{Y[n + 2] Y[n]} SVW 3 The prbability that a driver stps t pick up a hitchhiker is 004; different drivers f curse make their decisins t stp r nt independently Given that ur hitchhiker has cunted 30 cars passing her withut stpping what is the prbability that she will be picked up by the 37th car r befre? Hint this is a very simple prblem just clearly define yur assumptins
4 During the summer the number f vehicles passing a radside ice cream stre is mdeled by a Pissn prcess; vehicles cming frm the Nrth pass by at a rate f 60 vehicles per hur while vehicles cming frm the Suth pass at a rate f 80 per hur (Yu may assume that the traffic flw frm the tw directins are independent f each ther rrespective f directin f arrival assume that the vehicles visit the stre with prbability 01 and that they hld frm 1 t 5 passengers with prbabilities 03 03 02 01 and 01 respectively (Yu may assume that the cars are independent phenmena What is the average number f custmers per hur at the stre? What is the standard deviatin f the number f custmers? Hint mdel this prblem as a cmpund Pissn prcess as described in Chapter 22 (ie define what U \ n page 723 wuld need t be and use the results f that sectin and f prblem 2122 5 n general finding prbabilities fr crrelated Gaussian randm pairs (say X ] and X B is difficult The gal f this prblem is t wrk ut ne case f utility in many real prblems Let s assume that [X ] X B ]~N(0011 ρ fr simplicity assume that ρ > 0 and that we are interested in the prbability that bth variables are psitive; specifically the jint prbability Prb (X ] > 0 X B > 0 The slutin is a multistep prcess that want yu t prvide the details f: a First change variables t 1 0 Y ] d e ρ 1 Y B i1 ρ B i1 ρ B X ] d e X B What is the jint pdf f Y ] and Y B? b With this transfrmatin what is the regin f interest in the Y ] and Y B plane crrespnding t the event {X ] > 0 X B > 0}? Clearly sketch the answer c Cnvert frm variables Y ] and Y B t plar crdinates f magnitude R and phase θ What is the jint pdf f these variables? d Describe the regin f interest in terms f R and θ Hpefully it is nw clear hw t slve fr the desired prbability Write the prbability as a functin f ρ e Check that yur answer makes sense fr bth ρ 0 and ρ 1
1 2 Mz in thy 7 Elznzm EL ( Xntyn ( xmtym xnxm } t El Xnym t E ( Yntmlt E/YnYm 9 t m 't 12+12 t 16 t 3 (fu 49 t 3 ( T 'ṁ t ( f m S Cz Rzz ME 3 ( fl m 't ( ± 'M ml b yes 2 is Gauss sin and Wss and strict stat e Zun N ( 7 H plz a 9 QC Q( c 0159 d nthing changes
mtj 3 need fy Y is GRP & Wss My O Cy ( mm El E in A El Yn Yu k O re hn Xm ; hi a A EE huh ; f El in n km 57 j g ( n K j Mtk n a E hn hktm in min c cs k ' Cx 2 : E144 i! in :( t! E ' t a 's ' Cy Cat # kctskte
496 Pnh ' h general ( equate eg 1241 El 2147 get ( Y mil a EfYntilYn7ztYnbElYnnl7h_tyYn@Bernu1htnidspa04Pf37s 2 37 picks her up Came d ' 0248
23 Elk 3( 4 He custmers NCH Uk the 10% that stp stpping Passer with rate make the # f car 69% 14 than ( eg page 725 He cstumes cn as (1 t is G t 2 (3 t ( Elt (5 24 EL custmer 332 El NCH r P ;Eez Vr [ cstume 14 14 El @ shw a 3 ( 17 t ' P 235 24 tcz/t1(y4 tt C 54 714
' 5 H k 1 4 a : Ell ill : c : k : L :L y Yu n N (! b i : savin ' e
R c n rayleigh Q r units n C 0247 d angle is smile s 0<90 tante s P ( X > u X > u ZT t sine e e Yy f Ya