Continuous Random Variables: Basics

Transcription

1 Continuous Rndom Vrils: Bsics Brlin Chn Dprtmnt o Computr Scinc & Inormtion Enginring Ntionl Tiwn Norml Univrsit Rrnc: - D.. Brtss, J. N. Tsitsilis, Introduction to roilit, Sctions

2 Continuous Rndom Vrils Rndom vrils with continuous rng o possil vlus r quit common Th vlocit o vhicl Th tmprtur o d Th lood prssur o prson tc. Evnt { outcom } Smpl Spc Ω Evnt { outcoms } Evnt {c outcoms d} c d roilit-brlin Chn

3 roilit Dnsit Functions (/) A rndom vril is clld continuous i its proilit lw cn dscrid in trms o nonngtiv unction ( ), clld th proilit dnsit unction (DF) o, which stisis or vr sust B o th rl lin. ( B ) B d Th proilit tht th vlu o lls within n intrvl is ( ) d roilit-brlin Chn 3

4 roilit Dnsit Functions (/) Illustrtion o DF Evnt { outcom } Smpl Spc Ω Evnt {c outcoms d} Evnt { outcoms } Notic tht For n singl vlu, w hv d Including or cluding th ndpoints o n intrvl hs no ct on its proilit Normliztion proilit c d ( ) ( < ) ( < ) ( < < ) d ( < < ) roilit-brlin Chn 4

5 Intrprttion o th DF [ ] For n intrvl, + δ with vr smll lngth δ, w hv + δ ([, + δ ]) ( t ) dt ( t ) δ Thror, cn viwd s th proilit mss pr unit lngth nr is not th proilit o n prticulr vnt, it is lso not rstrictd to lss thn or qul to on roilit-brlin Chn 5

6 Continuous Uniorm Rndom Vril A rndom vril tht ts vlus in n intrvl [, ], nd ll suintrvls o th sm lngth r qull lil ( is uniorm or uniorml distriutd), i, othrwis Normliztion proprt d d roilit-brlin Chn 6

7 Rndom Vril with icwis Constnt DF Empl 3.. Alvin s driving tim to wor is twn 5 nd minuts i th d is sunn, nd twn nd 5 minuts i th d is rin, with ll tims ing qull lil in ch cs. Assum tht d is sunn with proilit /3 nd rin with proilit /3. Wht is th DF o th driving tim, viwd s rndom vril? c i 5, i 5, othrwis. 3 d 5 3 d 5 ( sunn d ) 5 ( rin d ) c, c,, 5, c 5 5 c c d d 5c 5c /5 /5 5 5 roilit-brlin Chn 7

8 Eponntil Rndom Vril An ponntil rndom vril hs DF o th orm, Normliztion roprt, i is positiv prmtr chrctrizing th DF, othrwis, d d Th proilit tht cds crtin vlu dcrss ponntill ( ) d roilit-brlin Chn 8

9 Norml (or Gussin) Rndom Vril A continuous rndom vril is sid to norml (or Gussin) i it hs DF o th orm ( μ ) σ, - π σ ll shp μ Whr th prmtrs nd r rspctivl its mn nd vrinc (to shown lttr on!) σ Normliztion roprt ( μ ) σ d π σ (?? S th nd o chptr prolms) roilit-brlin Chn 9

10 Normlit is rsrvd Linr Trnsormtions I is norml rndom vril with mn nd vrinc σ, nd i ( ) nd r sclrs, thn th rndom vril μ Y + is lso norml with mn nd vrinc E [ Y ] μ + ( Y ) σ vr roilit-brlin Chn

11 Stndrd Norml Rndom Vril A norml rndom vril with zro mn μ nd unit vrinc σ is sid to stndrd norml Y - Y ( ), π Normliztion roprt d π Th stndrd norml is smmtric round roilit-brlin Chn

12 Th DF o Rndom Vril Cn Aritrril Lrg Empl 3.3. A DF cn ritrril lrg. Considr rndom vril with DF Th DF vlu coms ininit lrg s pprochs zro Normliztion roprt, i <,, othrwis, ( ) d d roilit-brlin Chn

13 Epcttion o Continuous Rndom Vril (/) Lt continuous rndom vril with DF Th pcttion o is dind E [ ] ( ) Th pcttion o unction hs th orm g E g g d (?? S th nd o chptr prolms) Th vrinc o is dind vr d [ ] [ ] E[ ] ( ) E ( E[ ]) d W lso hv vr [ ] E E[ ] roilit-brlin Chn 3

14 Epcttion o Continuous Rndom Vril (/) Y + I, whr nd r givn sclrs, thn E [ Y ] E[ ] +, vr ( Y ) vr( ) roilit-brlin Chn 4

15 roilit-brlin Chn 5 Illustrtiv Empls (/3) Mn nd Vrinc o th Uniorm Rndom Vril othrwis, i, [ ] d d + E [ ] d + + E [ ] [ ] 3 vr E E

16 roilit-brlin Chn 6 Illustrtiv Empls (/3) Mn nd Vrinc o th Eponntil Rndom Vril othrwis,,, i, [ ] + d d d d d Q E [ ] [ ] + + d d d d d E E Q [ ] [ ] vr E E Intgrtion prts

17 roilit-brlin Chn 7 Illustrtiv Empls (3/3) Mn nd Vrinc o th Norml Rndom Vril -, σ μ π σ [ ] [ ] [ ] μ μ μ σ π π π σ μ + + -, Y Lt - - Y d Y Y E E E [ ] vr d d d Y Y π π π π E d d Q vr vr σ σ Y? (s Sc. 3.6)

18 Cumultiv Distriution Functions Th cumultiv distriution unction (CDF) o rndom vril is dnotd F nd provids th proilit F ( ) p ( ) () t i is discrt dt, i is continuous Th CDF F ccumults proilit up to Th CDF F provids uniid w to dscri ll inds o rndom vrils mthmticll, roilit-brlin Chn 8

19 roprtis o CDF (/3) Th CDF is monotonicll non-dcrsing F i i, thn j F i F j Th CDF tnds to s, nd to s F I is discrt, thn F is picwis constnt unction o roilit-brlin Chn 9

20 roilit-brlin Chn roprtis o CDF (/3) I is continuous, thn is continuous unction o F c c d c or c, dt t dt t F or, dt dt t F

21 roprtis o CDF (3/3) I is discrt nd ts intgr vlus, th MF nd th CDF cn otind rom ch othr summing or dirncing F p ( ) ( ) p ( i) i ( ) ( ) ( ) F ( ) F ( ) I is continuous, th DF nd th CDF cn otind rom ch othr intgrtion or dirntition F ( ) ( t ) df d Th scond qulit is vlid or thos or which th CDF hs drivtiv (.g., th picwis constnt rndom vril), dt, roilit-brlin Chn

22 An Illustrtiv Empl (/) Empl 3.6. Th Mimum o Svrl Rndom Vrils. You r llowd to t crtin tst thr tims, nd our inl scor will th mimum o th tst scors. Thus, m {, }, whr,, 3 r th thr tst scors nd is th inl scor Assum tht our scor in ch tst ts on o th vlus rom to with qul proilit /, indpndntl o th scors in othr tsts. Wht is th MF o th inl scor? p 3 Tric: comput irst th CDF nd thn th MF! roilit-brlin Chn

23 roilit-brlin Chn 3 An Illustrtiv Empl (/) ,, p F Q

24 CDF o th Stndrd Norml Th CDF o th stndrd norml Y, dnotd s Φ, is rcordd in tl nd is vr usul tool or clculting vrious proilitis, including norml vrils Φ ( Y ) ( Y < ) t / dt Th tl onl provids th vlu o Φ or Bcus th smmtr o th DF, th CDF t ngtiv vlus o cn computd orm corrsponding positiv ons Φ Y π (.5) ( Y.5) ( Y.5) Φ (.5) Φ or ( ) Φ( ) ll roilit-brlin Chn 4,

25 Tl o th CDF o Stndrd Norml roilit-brlin Chn 5

26 CDF Clcultion o th Norml Th CDF o norml rndom vril with mn nd vrinc σ is otind using th stndrd norml tl s ( ) Y Φ μ σ μ σ μ σ μ μ σ μ Lt Y. Sinc is norml nd Y is linr unction σ Y hnc is lso norml (with mn nd vrinc ). E [ ] [ ] μ vr ( ) E Y, vr Y σ σ o, roilit-brlin Chn 6

27 Illustrtiv Empls (/3) Empl Using th Norml Tl. Th nnul snowll t prticulr gogrphic loction is modld s norml rndom vril with mn o μ 6 inchs, nd stndrd dvition o σ. Wht is th proilit tht this r s snowll will t lst 8 inchs? 8 8 Y Φ () roilit-brlin Chn 7

28 Illustrtiv Empls (/3) Empl Signl Dtction. A inr mssg is trnsmittd s signl tht is ithr or +. Th communiction chnnl corrupts th trnsmission with dditiv norml nois with mn μ nd vrinc σ. Th rcivr concluds tht th signl (or +) ws trnsmittd i th vlu rcivd is < (or, rspctivl). Wht is th proilit o rror? Y N N + roilit-brlin Chn 8

29 roilit-brlin Chn 9 Illustrtiv Empls (3/3) roilit o rror whn snding signl - roilit o rror whn snding signl Φ Φ < < < + < σ σ σ σ N N N Φ σ σ σ N N N Y mn o N vrinc o N

30 Mor Fctors out Norml Th norml rndom vril pls n importnt rol in rod rng o proilistic modls It modls wll th dditiv ct o mn indpndnt ctors, in vrit o nginring, phsicl, nd sttisticl contts Th sum o lrg numr o indpndnt nd idnticll distriutd (not ncssril norml) rndom vrils hs n pproimtl norml CDF, rgrdlss o th CDF o th individul rndom vrils (S Chptr 7) W + + K + (, n n W cn pproimt n proilit distriution (th DF o rndom vril) with th linr comintion o n nough numr o norml distriutions Y ( ) α ( ) + α ( ) + K + α ( ) (, K, r norml, α K, n r i.i.d.) K ) roilit-brlin Chn 3

31 roilit-brlin Chn 3 Rltion twn th Gomtric nd Eponntil (/) Th CDF o th gomtric Th CDF o th ponntil Compr th ov two CDFs nd lt,,k or go n p p p p p p n F n n n or p > d F δ δ n p p n p p n p > or ln lt ln δ δ

32 Rltion twn th Gomtric nd Eponntil (/) F δn p ( δn) ( p) F ( n) n go roilit-brlin Chn 3

33 Rcittion SECTION 3. Continuous Rndom Vrils nd DFs rolms, 3, 4 SECTION 3. Cumultiv Distriution Functions rolms 6, 7, 8 SECTION 3.3 Norml Rndom Vrils rolms 9,, roilit-brlin Chn 33