Department of Mathematics, IST Probability and Statistics Unit Probability Theory. Group I Independence. Group II Expectation W (5) = 0 N(5)

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1 Dertmet of Mthemtics, IST Probbility d Sttistics Uit Probbility Theory d. Test Recurso ) st. Semester 4/5 Durtio: h3m 5//4 9:45 AM, Room P Plese justify your swers. This test hs two ges d three grous. The totl of oits is.. Grou I Ideedece 3.5 oits The wter of certi reservoir is deleted t costt rte of uits dily. The reservoir is refilled by rdomly occurrig riflls; they occur ccordig to Poisso rocess with rte. er dy; d the mout of wter dded to the reservoir by rifll is 5 uits with robbility.8 or 8 uits with robbility.. The reset wter level is t 5 uits. ) Defie i some detil the r.v. W t) reresetigthewterlevelofthereservoirttime.) t. Auxiliry r.v. d stochstic rocess i wterrefillbythei th rifll i, i 8 N ><.8, x 5 P x)., x 8 >:, x 5, 8 Nt) umberofrifllsutotimeidys)t {Nt), t } PP.) d ideedet of the i Nt) Poisso t.t) Defiig W t) Sice the wter of the reservoir is deleted t costt rte of uits dily, etc., the wter level of the reservoir t time t is give by 8 9 < Nt) W t) mx :, 5 t + i ; curret level wter deleted + wter from the riflls). i b) Fid the robbility tht the reservoir will be emty withi the ext five dys?.) Evet Citlizig o the fct tht the r.v. i re ositive, the reservoir will be emty withi the ext five dys i Note: P i i ; P Nt) i i is comoud PP. W 5) N5) 5 5+ i le i N5) i le i N5), tht is, i it does ot ri i those 5 dys. Requested robbility P [W 5) ] P [N5) ] N5) Poisso. 5) F Poisso) ) tbles.379. c) Wht is the robbility tht the reservoir will be emty withi the ext te dys?.5) Evet Cosiderig tht the r.v. i tke vlues 5 d 8, the reservoir will be emty withi the ext te dys i W ) N) 5 + i le Requested robbility P [W ) ] Grou II Execttio i N) i i le 5 N) or N) d 5). N)?? i P [N) ] + P [N) ] P 5) N) Poisso. ) F Poisso) ) + [F Poisso) ) F Poisso) )].8 tbles ) oits. Let: Uiform, ); <<b<; Y I,b) ); Z ) I,) ). ) Show tht Y,Z L d yet Y Z L..) Uiform, ) Y I,b) ), <<b <b<), otherwise

2 Z ) I,) ) Obs.: <<b<. To rove Y,Z L Proof Sice <, b <, E Y ) E[ I,b) ) ] Cor.4.8 Uiform,) ), < < <<), otherwise Z + x I,b) x) f x) dx x dx lx) b + E Z ) E[ ) I,) ) ] Cor.4.8 Uiform,) Z + Z x) I,) x) f x) dx x dx l x) +, i.e., Y d Z re ot itegrble r.v. Y,Z L ). Checkig whether Y Z L Recllig tht <<b<, we obti E Y Z ) E[ I,b) ) ) I,) ) ] Cor.4.8 Uiform,) Z + x I,b) x) x) I,) x) f x) dx x x) dx x + dx x lx) b l x) b lb/)+l[ )/ b)] < +, tht is, Y Z is itegrble r.v. Y Z L ). b) Does the revious exmle cotrdict Cuchy-Schwrz s iequlity? Justify your swer..5) Commet Cuchy-Schwrz s iequlity rovides su ciet coditios o r.v. Y d Z to be delig with itegrble roduct Y Z: we must be delig with Y,Z L. 3 However, i the revious exmle we hve Y,Z L d therefore Y,Z L,by Lyuov s iequlity L L ), therefore we cot eve ly such iequlity let loe cotrdict it.. ) Prove from scrtch!) tht if is bsolutely cotiuous d oegtive r.v. i.5) L the P ) le E), for >. To rove bsolutely cotiuous,, L ) P ) le E), for > Proof P ) x, >,x/ le xf x) le Z + Z + Z + E). f x) dx x f x) dx xf x) dx b) Now, dmit tht L..) For which vlues of c the revious uer boud be surely imroved? Aother uer boud for P ) By ivokig Mrkov s iequlity with,weget,> P ) P ) le E ). Comrig the uer bouds By tkig dvtge of the chrcteristics of the r.v., we c dd tht the ltter boud is stricter th the oe we obtied i ) for : E ) < E) > E ) E). 3. Let,,... be r.v. to, whose.d.f. is f x) e x,x R. Suose we.5) cotiue smlig from this oultio util egtive observtio ers. Let S be the sum of the observtios thus obtied icludig the egtive oe). Obti the exected vlue d the vrice of S. Hit: Recll tht ES) E[ES N)] d V S) V [ES N)]+E[V S N)], where N is the rdom umber of observtios collected icludig the egtive oe). d commo.d.f., exected vlue d vrice i, i N 4

3 f x) e x,x R Moreover, sice the.d.f. is symmetric roud the origi, Other r.v. E) V ) E ) Cor.4.8 Z + Z + Z Z + Z + x 3.45) 3) 3 )!. x f x) dx x e x dx x ex dx + x e x dx e x dx Z + x e x dx N umber of observtios collected util egtive observtio ers icludig the egtive oe) N Geometric), where P <) becuse the.d.f. is symmetric roud the origi. S P N i i sumoftheobservtiosthusobtiedicludigtheegtiveoe) Requested exected vlue d vrice [Tkig ito ccout tht N is ideedet of the i,] ES N) is r.v. tht tkes vlue! ES N ) E i i E) with robbility, for y N. Furthermore, V S N) isr.v.thttkesvlue! V S N ) V i i V ), with robbility P N ) ), N. As cosequece, ES) E[ES N)] 5 V S) V [ES N)] + E[V S N)] V ) + EN) N Geo) / Visitors of turl rk choose t rdom oe out of ths d do it ideedet of oe other. ) Fid the robbility tht ths d re chose twice ech, ths 3, 4 d 5 re.5) chose oce ech, d th is ot chose t ll by the 7 visitors. Rdom vector N N,...,N ) N i umber of times th i is chose by visitors, i,..., Distributio N Multiomil d, ) Prmeters d 7,..., )/,...,/) becuse the ths re chose t rdom) Joit.f. P N ) P N,...,N ) Def ! Y i Q i, i! for i {,,...,7}, i,...,, such tht P i i 7. Requested robbility P N,N,N 3,N 4,N 5,N ) 7!!!!!!! '.45. i b) Obti the exected vlue d vrice of the umber of visitors who choose th.) give tht 3 of the 7 visitors chose th. N N ) Rem. 4.3 Biomil, Requested exected vlue d vrice EN N 5) Rem. 4.3 or P ro ) '.8 V N N 5) 7 3) '.4

4 Grou III Covergece of sequeces of r.v..5 oits. The Preto distributio, med fter the Itli ecoomist Vilfredo Preto, ws origilly 3.) used to model the welth of idividuls,. We sy tht Pretob, ) if f x) b,x b, x + where b>isthemiimumossiblevlueof d > isclledthepretoidex.let: {,,...} be sequece of idividul welths with commo Pretob, ) distributio; Y mx i,..., i be the scled mximum welth of the first idividuls. Show tht {Y,Y,...} coverges i distributio to Y, whose c.d.f. is give by F Y y) e y,y>. i, i IN Preto, ), > Commo.d.f. d c.d.f. f x), x <, x x + F x), x < R x dt t x x, x Rge R [, +) Aother r.v. Y mx i,..., i ) Rge h R Y, + R Y! R +,s! + C.d.f. For y, h i F Y y) P ) le y i P h ) le y i hf y Exmle3.7 le y y. Checkig the covergece i distributio Sice i) lim!+ F Y y) lim!+ + y e y,y>, ii) F Y y) e y,y>, is the c.d.f. of the bsolutely cotiuous r.v. Y, 3 iii) lim!+ F Y y) F Y y), for ll the cotiuity oits of the c.d.f. of Y, we c coclude tht Y d! Y.. Bsed o iformtio from revious study, grou of orithologists believe tht roximtely 55% of the red til hwk oultio cosists of femle hwks. ) Wht is the reltive error if we ly the DeMoivre-Llce locl limit theorem.5) to roximte the robbility tht exctly red til hwks re femles if the orithologists smle 4 red til hwks? Commet., if the i th red til hwk is femle i, otherwise i, i,,..., 4) Beroulli.55) Aother r.v. S P i i umberoffemlesismleofsize red til hwks S Biomil, ) P S s) s s ) s,s,,..., Requested robbility 4 P S 4 ).55 s 4.55).54. Aroximte vlue Assumig the DeMoivre-Llce locl limit theorem Theorem 5.77) c be lied the P S 4 ) 5.8) ' ) ) 8 < ex : ' The footote leds us to believe tht Y Fréchet ) this is ideed the cse!). # ) # ) ; I.e., the Preto distributio belogs to the domi of ttrctio of the Fréchet distributio. 7 8

5 Reltive error d commet It is give by %.3797% d rther smll, suggestig tht the roximtio followig from the DeMoivre- Llce locl limit theorem is very resoble i this cse. b) Fid the miiml smle size so tht the smle will coti more th 5% femles,.) with robbility of t lest 9%. Hit: It my be useful to ivoke the Lideberg-Lévy Cetrl Limit Theorem., if the i th red til hwk is femle i, otherwise i, i,,... E) µ.55 V ) ) < Aother r.v. S P i i umberoffemlesismleofsize red til hwks Aroximte distributio of S Let {Z,Z,...} be the sequece of the stdrdized rtil sums, where S ES) S µ Z. The, ccordig to the Lideberg-Lévy CLT, Z d! V S) Norml, ). Moreover, for su cietly lrge vlue of, s µ P S le s) '. Requested umber This umber c be obtied roximtely: N : P S >.5).9 #.5 ).9.9).5 ).9) ).8 #.55.55) , hece, the miiml le smle size is [Note tht ' ] ) 9