Avd. Matematisk statistik

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1 Avd. Mtemtisk sttistik TENTAMEN I SF94 SANNOLIKHETSTEORI/EAM IN SF94 PROBABILITY THE- ORY WEDNESDAY 8th OCTOBER 5, 8-3 hrs Exmitor : Timo Koski, tel , emil: tjtkoski@kth.se Tillåt hjälpmedel Mes of ssistce permitted : Appedix i A.Gut: A Itermedite Course i Probbility. Formuls for probbility theory SF94. L. Råde & B. Westergre: Mthemtics Hdbook for Sciece d Egieerig. Pocket clcultor. You should defie d expli your ottio. Your computtios d your lie of resoig should be writte dow so tht they re esy to follow. Numericl vlues should be give with the precisio of two deciml poits. You my pply results stted i prt of exm questio to other prt of the exm questio eve if you hve ot solved the first prt. The umber of exm questios Uppgift is six 6. Solutios writte i Swedish re, of course, welcome. Ech questio gives mximum te poits. 3 poits will gurtee pssig result. The grde Fx the exm c completed by extr exmitio for those with 7 9 poits. Solutios to the exm questios will be vilble t strtig from Wedesdy 8th OCTOBER 5 t 5.3. The exm results will be ouced t the ltest o Tuesdy the th of November, 5. Your grded exm pper c be retied t the Studet ffirs office of the Deprtmet of Mthemtics durig period of seve weeks fter the dte of the exm. Lyck till!

2 forts tetme i sf Uppgift, Y is bivrite r.v. with the joit p.d.f. { xe f,y x, y x xy, for x >, y >, elsewhere. Determie the distributio of + Y. p Uppgift N, σ x d fx is the p.d.f. of N, σ c. U U, f d is idepedet of. Fid the probbility b Fid the probbility P { x} {U f}. P U f. 4 p 4 p c Estblish by the precedig tht U f N, s, where +. p s σc Uppgift 3 U hs symmetric Beroulli distributio, i.e., its p.m.f. is / if k, p U k / if k +, otherwise. V Exp d is idepedet of U. Show tht UV L. b i, i,,..., re I.I.D., i L. Show tht i i χ. 5 p 5 p

3 forts tetme i sf Uppgift 4 { } is sequece of rdom vribles with vlues i the itervl [, ]. We set F σ,,...,. We ssume tht, where < <. Let us lso ssume tht for,,... P + F, d P + + F. We kow i view of sf94 Homework 5 tht {, F } is mrtigle. You will eed this fct, but You re ot required prove this piece of kowledge ew. Show ow isted tht E [ + ] 4 E [ ]. 4 p b It c be show tht, s +, plese tke this covergece for grted here. Prove tht the limitig r.v. stisfies E [ ]. 3 p c Fid the distributio of from the equlity stted i Uppgift 4 prt b bove. You re expected to preset your rgumets i the precise detil. Aid: It holds for mrtigles tht E [ ] E [ ] for ll. 3 p Uppgift 5 Let {t < t < } be wekly sttiory process with me fuctio tht equls µ, d with the utocovrice fuctio Cov h e h, >. Show tht for y t t + P t, s +. p

4 forts tetme i sf Uppgift 6 W {W t t } is Wieer process. We defie the joitly Gussi sequece.k.. process i discrete time of bivrite r.v. s k, Y k by mes of k def cosπktdw t, k,,..., d Fid Y k def siπktdw t, k,,.... Cov k, l, Cov Y k, Y l, Cov k, Y l for ll vlues of k d l, where k, l. p b Expli why k k k k k k N, π, s +, 6 d why k k Y k d k k k. 3 p c Show tht k k k k + k Y k d N,, s +. Determie explicitly the vlue of. You re expected to justify your solutio crefully. 5 p

5 Avd. Mtemtisk sttistik SOLUTIONS TO THE EAM WEDNESDAY THE 8 th OF OCTOBER, 5. Set Here U is clerly uxiliry vrible. The Sice Y V U Hece J u gives i.e., the desired p.d.f is or Uppgift U, V + Y. h U, V U, Y h U, V V U., it holds tht V U. We eed the Jcobi determit x x J u v y u u u y v v u. d isertio i the formul, foud i the Formelsmlig, f U,V u, v f,y h u, v, h u, v J, f U,V u, v e v, v u, f V v f V v v e v du ve v, v >. { ve v, for v >, elsewhere. A serch i the Tble of Appedix B shows tht V Γ,. ANSWER : + Y Γ,. Uppgift The desired probbility c be writte s P { x} {U f} d sice U d re idepedet, ft f U uf tdudt ft ft f U, u, tdudt f U uduf tdt

6 forts tetme i sf sice U, f, i.e., PU x σ c π. Hece σ c π e t /σc ft f f tdt, x. We hve ft f σ c π e t /σc d f e t / f tdt σ c π t /σ e x dt σ x π e t σ + c dt. σ x π It looks like this is for our purposes the most suitble wy to express the desired probbility. ANSWER : P { x} {U f} t x σ x π e b The probbility here is doe s bove with smll modifictio: P U f + ft σc + f U uduf tdt + σ c π e t / t /σ e x dt σ c π σ x π σ x π + σ x π By the properties of the p.d.f. of N, + e t ANSWER b: P U f σ x e t /σ c e t /σ x dt + e t σc + σc + σ c σc + + c By the defiitio of coditiol probbility P { x} {U f} dt we hve Whe we isert the swers from d b, we obti t x σ x π e. σ x σ c + σ x dt. π. dt. P { x} {U f}. P U f σ c σc + + dt

7 forts tetme i sf π σ c + x e t σc + dt. But the ltest expressio is P Y x, whe Y N, s, where s σ c Uppgift 3 This problem higes upo keepig i mid tht if V Exp, the Tke y >. The +. PV. P UV y P V y U P U + P V y U + P U + P V y + P V y, s U hs symmetric Beroulli distributio d is idepedet of V. By simple rithmetic P V y + P V y + e y/, sice V Exp d, i view of, we get P V y for y >. Thus the p.d.f. of UV is for y > f UV y d P UV y dy e y/ For y < we get s bove P UV y P V y + P V y, d by we hve P V y for y <. The we hve, s y >, d sice F V y e y/ for y >, Thus the p.d.f. of UV is for y < P V y F V y ey/. f UV y d P UV y dy ey/ If we write the two results bout f UV y bove s sigle expressio, we get clerly This suffices to show tht s ws climed. f UV y e y /, UV L, < y < +.

8 forts tetme i sf b If i, i,,..., re I.I.D., i L, the the prt of this Uppgift shows tht there re I.I.D. U i symmetric Beroulli distributio, d I.I.D. V i Exp, V s d Us idepedet, d such tht Thus it follows We compute the chrcteristic fuctio of Sice V i re I.I.D., the i d U i V i, i,,...,. i d V i, i,,...,. i i ϕ i where V Exp. Furthermore, ϕ V i V i t ϕ V We fid from our resources i Bilg B tht Thus we obti V i. t, t ϕ V t. ϕ V t it. V ϕ V t i t From Bilg B we observe ow tht it /. it is the chrcteristic fuctio of χ. Hece it follows by uiqueess of chrcteristic fuctios tht i i χ, s ws climed. Uppgift 4 We strt by the double expecttio E [ + ] E [ E [ + ]] F. I the ier expecttio E [ + ] [ ] F E + F E [+ F ] + E [ ] F.

9 forts tetme i sf I the middle term we first tke out wht is kow d get d sice {, F } is mrtigle, E [ + F ] E [ + F ],. I the lst term we tke gi out wht is kow d fid E [ F ] E [ F ]. Filly, by the lw of the ucoscious sttistici, the first term is E [ ] + F Whe we collect the results bove we hve the ier expecttio s E [ + F ] Hece which equls which is s desired E [ + ] E 4 E [ ], [ ] 4, 4 b The fct tht, s +, is istce of the mrtigle covergece theorem, tke for grted here, d proved i my textbooks i dvced probbility theory. Covergece implies, s +, E [ ] E [], E [ + ] E [ ] d Hece Furthermore Therefore we get E [ ] E [ ]. E [ + ] E [ +] E [+ ] + E [ E [ ] E [ ] + E [ ]. 4 E [ ] E [ ]. 4 E [ ]. ]

10 forts tetme i sf c Set Y. Sice with probbility oe, the. I sf94 Homework 5 it ws estblished tht if P, s +, d P, the P. Sice here s +, it lso holds tht P s +. By we hve tht Y, with probbility oe. I dditio, we hve show tht E [Y ]. But, if the o egtive rdom vrible Y hs zero s expecttio, the the vrible Y is with probbility oe. Thus with probbility oe. But the oly rel umbers stisfyig x x re x d x. Hece the r.v. hs oly these two vlues, d. We kow tht mrtigle hs costt me, d therefore E [] lim E [ ] lim E [ ] E [ ]. + + O the other hd, s hs oly two vlues, d, E [] P + P P. Hece we hve Thus ANSWER c: Ber. P. Uppgift 5 If {t < t < } is wekly sttiory process with me fuctio µ, the the utocovrice fuctio is E [t µ s µ] Cov t s e t s. We show first tht for y t t + t, s +. We cosider E [ t + ] [ t E t + ] µ t µ

11 forts tetme i sf E [ t + ] [ µ E t + ] µ t µ + E [ t µ ]. By the bove E [ t + ] µ Cov, E [ t µ ] Cov, d E [ t + ] µ t µ Cov e. Hece E [ As +, d >, t + ] t e + e e., d therefore we hve show tht t + t, s +. But this implies tht t + P t, s +, s ws to be show. Uppgift 6 By the properties of the Wieer itegrl c.f., Formelsmlig E [ k ] E [Y k ]. Furthermore, by the properties of the Wieer itegrl, s covered i the Formelsmlig, { k l E [ k l ] cosπkt cosπltdt k l, { k l E [Y k Y l ] siπkt siπltdt k l, d E [ k Y l ] cosπkt siπltdt for ll k, l. These results c be foud with itegrtio by prts, d re well kow i mthemtics Fourier series, see L. Råde & B. Westergre: Mthemtics Hdbook for Sciece d Egieerig, chpter 3, sectio 3., p. 3. Thus { k l Cov k, l k l,

12 forts tetme i sf d { k l Cov Y k, Y l k l, Cov k, Y l. b Sice k, Y k is joitly Gussi sequece of r.v. s, it holds tht i view of tht k re I.I.D. vribles d by properties of the Wieer itegrl, k N,. Note tht Vr k Cov k, k d VrY l Cov Y l, Y l. The sum k coverges i me squre if d oly if + k k k k < + By L. Råde & B. Westergre: Mthemtics Hdbook for Sciece d Egieerig, chpter 8, sectio 8.6, p.94 we hve The sum + k k π 8. k k k is i dditio Gussi r.v.. Hece it follows by properties of me squre covergece of sums of I.I.D. orml r.v.s k N, tht k k k N, π, 6 s +. The vrice of the limit is Vr[ ] + k k The sme rgumet is clerly vlid for k Y k k d hece k k Y k N, π. 6 π 8.

13 forts tetme i sf c We write k k k k + k Y k / / k k k k + /. k Y k By the Cetrl Limit Theorem, s k N, re I.I.D. vribles, we get tht / d k N,, k s +. Becuse k N, re I.I.D. vribles, the Wek Lw of Lrge umbers yields k P E [ ], k s +. Becuse Y k N, re I.I.D. vribles, the Wek Lw of Lrge umbers yields Y P k E [ ] Y, k s +. It holds lso by theorem i the course/ln tht the sum of the two sums bove coverges i probbility to costt k + k k Y k P + s +. The the Crmér-Slutzky theorem gurtees tht k k k k + d k Y N,. k