) and furthermore all X. The definition of the term stationary requires that the distribution fulfills the condition:

Transcription

1 Assigm Thomas Aam, Spha Brumm, Haik Lor May 6 h, 3 8 h smsr, 357, 7544, 757 oblm For R b X a raom variabl havig ormal isribuio wih ma µ a variac σ (his is wri as ~ (,) X. by: R a. Is X ) a urhrmor all X ar ip. W i h sochasic procss saioary? R b. Has ip icrms? R c. Has saioary icrms? R. Fi E [ ] a [ ] Var. Th iiio o h rm saioary rquirs ha h isribuio ulills h coiio: F ( ) ( )( x,..., x ) F ( s ),..., ( s )( x x ),...,,..., or all, s R. For h xprssio bcoms v mor hay: F I R ( )( x) F ( s )( x) is saioary h i shoul b ru ha: [ x] [ s x] [ X x] [ X s s x] [ (,) x ] [ (,) x s] Uorualy, h las li ors a coraicio. Thror w prov ha is o saioary. For ip icrms h raom variabls X ( ) X, X ( ) X,... ( ) X ( ) ip or all < <... < < ). Th irs sp is: R X ar a i T (ur h assumpio ha all i ar i ascig orr which mas X ( X ) s s X s For ay subsqu icrm: X s s X ( X r) s r r s r X s s X s r ormac Evaluaio Tchiqus summr rm 3

2 Assigm Thomas Aam, Spha Brumm, Haik Lor May 6 h, 3 8 h smsr, 357, 7544, 757 I was sa ha all X ar ip. Thror, all icrms X X, X X, c. hav o b,, c. ar ip as wll. Tha rsul las o h coclusio ha ip, oo. Bcaus s is a cosa valu i os o iluc ay pcy a hus has ip icrms. R Saioary icrms o accomplish: Rplacig s s ( s) ( s) ( ) by X yils: ( s) ( s) s s s ( s,) ( s,) (, ) (,) (,) ( ) ( ) Hrby w hav show proucs saioary icrms. R Th xpcaio valu is giv by E[ ] E[ X ] EX (rmmbr: ~ (,) Var. Furhrmor X ) ormac Evaluaio Tchiqus summr rm 3

3 Assigm Thomas Aam, Spha Brumm, Haik Lor May 6 h, 3 8 h smsr, 357, 7544, 757 oblm B ( ( ) ) R a Poisso ocss wih ra. W ar giv ha o arrival occurr i h irval (,] (hc ( ) ) bu w o o kow wh his arrival happ. L o h raom variabl o h irs arrival im, h rag o is (,]. Show ha has a uiorm isribuio (hc, is isribuio ucio is giv by F ( y) [ y] ). Hi: compu y ( ) y. A irsig propry o h Poisso procss is: [ ( ) k] ( ) k k Th law o coiioal probabiliis givs: [ ] [ y, ( ) ] y [ ( ) ] [ ( ) ] [ ( y) ] [ ( ) ] Applyig h ormula vlop abov givs: y y y ( y) y y y I coclusio, w show ha i ( ) occurs i h irval (,] h is uiormly isribu. ormac Evaluaio Tchiqus 3 summr rm 3

4 Assigm Thomas Aam, Spha Brumm, Haik Lor May 6 h, 3 8 h smsr, 357, 7544, 757 oblm 3 B ( ( ) ) R ip o ( ( ) ) wih siy ucio ( y) a Poisso ocss wih ra, a a coiuous o-gaiv raom variabl R. ( ) a. Show ha ( )( ) E ca b xprss as ( )( ) M ( ( ) ) Trasorm o ( ), i.. h - ca b xprss i rms o h mom graig ucio or. Hi: law o oal probabiliy or coiuous raom variabls. b. Us ( )( ) o i E [ ( )] a [ ( )] Var. Obviously: ( ) ( )( ) E Tha ca b xprss as: b [ ( ) ] [ ( ) ] ( ) Bcaus o ( x) x ( x) a a b x [ ] ( ) [ ( ) ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ormac Evaluaio Tchiqus 4 summr rm 3

5 Assigm Thomas Aam, Spha Brumm, Haik Lor May 6 h, 3 8 h smsr, 357, 7544, 757 ormac Evaluaio Tchiqus 5 summr rm 3 Accorig o Taylor s obsrvaios h ollowig hols: x x. Hc, w g: M Th compuaio o h xpcaio valu E yils: Th irs rivaio o a sum las o: Usig ha ormula givs: Tha igral ca b us o rmi h xpcaio valu: E E

6 Assigm Thomas Aam, Spha Brumm, Haik Lor May 6 h, 3 8 h smsr, 357, 7544, 757 ormac Evaluaio Tchiqus 6 summr rm 3 Th x sp, calculaig Var, ivolvs h uiliaio o: Var Agai, som rivaios hav o b o: W i: A ially: Var E E E E E E E Var

7 Assigm Thomas Aam, Spha Brumm, Haik Lor May 6 h, 3 8 h smsr, 357, 7544, 757 oblm 4 Somims o is irs i makig pricios,.g. w kow ha a il rasr alray ook im uis, wha is h probabiliy ha i aks aohr > im uis? A raom variabl X is call havy-ail, X > ~ or < <. A ucio ( x) bhavs asympoically as g ( x), wri ( x) x ~ g x, i hr xiss som c R, c such ha lim c hols, g ( x) rquir. x g x wh as a. Show ha h xpoial isribuio wih paramr is o havy-ail. b. Compu [ X > X > ] lim or h xpoial isribuio. c. Compu [ X > X > ] lim or h havy-ail isribuio.. Compar a irpr h rsuls I may suis i has b ou ha h isribuio o il sis i a UIX il sysm a h ocum sis o a wb srvr ar havy-ail. Th xpoial isribuio is rmi by: [ Exp( ) ] x x x Bcaus o [ X x] [ X > x] I ( x) g( x) : x [ Exp( ) > ] ( ) x ~ w hav o i som ( x) x lim c whr ( x) a ( x) x x g( x) g : lim lim lim ( ) sic is ix. As a rsul, hr os o xis som havy-ail, oo. c a hc h xpoial isribu is o ormac Evaluaio Tchiqus 7 summr rm 3

8 Assigm Thomas Aam, Spha Brumm, Haik Lor May 6 h, 3 8 h smsr, 357, 7544, 757 A vry hay propry o h xpoial isribuio, is mmorylssss, allows us o rmi h probabiliy i jus hr lis: lim [ Exp( ) > Exp( ) > ] lim [ Exp( ) > ] [ Exp( ) > ] For h havy-ail isribuio: lim [ X > X > ] lim lim lim [ X >, X > ] [ X > ] [ X > ] [ X > ] ( ) lim W i o hav h im o rasla our coclusios hr ar sill wri i rma: Für i Expoialvrilug imm i Wahrschilichki aür, ass i Übrragugsvorgag och i Zispa ahäl, uabhägig vom brach Zipuk, währ r Übrragug xpoill mi r Läg r Zispa ab. Dami wir i kur Zispa avorisir, uabhägig vo r scho vrsrich Zi. Das bu, ass as E r Übrragug immr höchswahrschilich ah is. Für i havy-ail-vrilug is i Wahrschilichki aür, ass i Übrragugsvorgag ahäl, immr ah Eis, uabhägig vom brach Zipuk u r Läg s Zisrs. Somi wir vorhrgsag, ass i Übrragug immr wir aaur u as E r Übrragug hr uwahrschilich is. Kur Daiübrragug lass sich rch gu mi r Expoialvrilug bschrib, lag Daiübrragug sprch agg hr ir havy-ail-vrilug. ormac Evaluaio Tchiqus 8 summr rm 3

9 Assigm Thomas Aam, Spha Brumm, Haik Lor May 6 h, 3 8 h smsr, 357, 7544, 757 oblm 5 (bous) Th isribuio a siy o a raom variabl X wih Paro isribuio ar giv by: F ( x) [ X x] ( x) k, x k x or > k > a k raom variabl wih paramrs paramr x. For < < h Paro isribuio is havy-ail. B X such a Paro, k. Furhrmor, b a xpoial raom variabl wih X > x > x or x usig a oubly-logarihmic scal.. Plo boh a Paro isribuio Expoial isribuio ormac Evaluaio Tchiqus 9 summr rm 3