Repeated Trials: We perform both experiments. Our space now is: Hence: We now can define a Cartesian Product Space.

Transcription

1 Rpatd Trals: As w hav lood at t, th thory of probablty dals wth outcoms of sgl xprmts. I th applcatos o s usually trstd two or mor xprmts or rpatd prformac or th sam xprmt. I ordr to aalyz such problms w must cocv a combd xprmt. Suppos w had two dpdt xprmts, E ad E : ) Rollg of a far d; S f, f, f, f, f, f whr { 5 6} 6 { f } { }... f 6 lmt ) Tossg of a co S H, T lmts { } ( H ) ( T ) S ad W prform both xprmts. Our spac ow s: { f h f h,..., f t, f t... },, ( f h) ( f ) ( h) lmts 6 S ad S d ot b dffrt Spacs. Thy could b th sam xprmt. For xampl, tossg of two far cos: {, } {, } { HH, HT, TH, TT} S H T S H T S [ HH] ad [ HH] [ H] [ H] Gv sts Hc: W ow ca df a Cartsa roduct Spac. S, S, S,..., S. Thr Cartsa roduct s dfd as th st * S * S *... S f f... f, S S *, whos lmts ar all ordrd tupls whr f s a lmt of th st S Smlarly, f w ar, gv xprmts, ε ( S, F, ),... Th combd xprmt s dfd as: ε ε ε * ε...* ε ( S, F ) Ad ts vts ar of th form a S * ε,. a * a * a *...* a, whr,

2 Not: f th xprmts ar dpdt, th ( a a * a *...* a ) ( a ) ( a ) ( a ) ( a )... * Lt us cosdr som xampls: Exampl { St of all ral umbrs x} { all ral x} { St of all ral umbrs y} { all ral y} S { st of all po t s x y plas} S S S S * Y X a) Th st S { x x x } { y y y } Y Y Y X X X S x x x S b) Th st { } Y X X X ad S S { y y y } Y Y Y S x x S fally { } X Y X X

3 Cosdr ow, tossg a co tms. Our combd xprmt s: ε ε * ε * ε *...* ε whr ε ε ε... ε & S S * S *...* S S lmts our spac composd of pot f,, whr f I s H or T., A vt would b [ f f,..., ] f Sc xprmts ar dpdt,, a mmbr of th Fld.. [ f f,..., f ] [ f ] [ f ] [ f ] [ ], f... whr [ f ] p f f H q or f T s th probablty of gttg a had or a tal ad p + q Th, K hads [,,,... ] a spcfc H H H T T ordr + Ths s th probablty that K Hads a spcfc ordr.... H H H H... H T or T T... H... H tc tc. p q --Gomtrc Dstrbuto, or Law Th probablty of all combatos of K hads s th sum of all combatos. j [ K hads toss] ( had ) whr j Numbr of combatos To dtrm ths quatty, w cosdr th followg; th probablty of a partcular squc of K had tosss [ K hads a partcular squc] p q, tosss. Ad ow w ow that thr!!( )! ar combato or K hads tosss [ K hads tosss] p q Bomal probablty law Ths ar calld Broull Trals, ad gral ar dfd as a squc of A dpdt rpttos of a radom xprmt, whr th xprmt has two outcoms.

4 ( K ) { a occurs K tms} p * q whr p probablty of a succss { A} q probablty of a falur { A} p+q W ca ow gralz ad df Gralz Broull as a squc of dpdt rpttos of a radom xprmt, wth two or mor outcoms...,, l, m ( l, m), l m, whr +l+m + + Gralzd Broull Trals- dpdt trals, ad o ach tral th rsult s o of possblts p, p,..., p, rspctvly: b b,..., b, wth probablty, p 0 S K whr, ( b b,..., b, b, b,..., b,..., ) s, b,..., b x th p () s p p... p x robablty of ay outcom such that thr ar occurrcs of b for,...,. Lt B,,..., " occurrcs of b for,...,!...!!...! (,,..., ) ( B B ),..., (wthout drvato)

5 5 Exampl A computr systm has thr I/O dvcs; a ds, a prtr ad a trmal. Each job for th systm rqusts xactly o I/O dvc, th ds wth probablty /, th prtr wth probablty 0., ad th trmal wth probablty 0.. W obsrv 0 dpdt rqusts for th I/O dvcs. ) Fd probablty that thr wll b 0 rqusts for ach dvc. ) Fd th probablty that thr wll b rqusts for th prtr. Soluto: ) Ths s a Gralzd Broull Tral wth 0,, 0, p 0.5, p 0., p 0. : / p 0! 0!0!0! ( 0,0,0) 0.5 * 0. * ) Collaps rqusts to ds ad to trmal togthr: ordary Broull tral. 0 8 (" prtr rqusts") * * Exampl: A far d s rolld 5 tms; ) Fd ( ) p that sx wll com up twc. 5 Lt a { f } { sx} 6 {} a 6 {} a {} a 5 5, ( a) [ 6 occurs tms] 5! ( 5 ) 5! 5!! ! 5! 6 6

6 6 Exampl: Th capta of a Navy guboat ordrs a volly of 5 mssls to b frd at radom alog a 500-foot strtch of shorl that h hops to stablsh as a bach-had. Dug to th bach s a 0-foot-log bur srvg as th my's frst l of dfs. What s th probablty that xactly thr shlls wll ht th bur? { }!!! ( 0.06) ( 0.9) 5*8* ( 0.06) ( 0.9) Exampl If a famly has four chldr, s t mor lly thy wll hav two boys ad two grls or thr of o sx ad o of th othr? Assum that th probablty of ay chld bg bor s /. Two of Each:!!! Thr of o, ad o of th othr: * * * * * 6 x W ca ow tur our attto to othr probablty laws. Cosdr, f w hav a ur wth N chps of whch r ar rd, w ar wht,.. r+wn Suppos w rach ad grab a tr sampl of sz. Crta of thm wll b rd Y. Lts fd th probablty that y s qual to a spcfd umbr, say. Wll, wth N objct, w ca gt;

7 7 N total combatos of sampls(wthout ordr) Also, f w hav r rd, ad w wat to b sampl, th combatos o r For ach of ths (abov) combatos, w hav w rmag for wht to fll out th sampl. Hc: ( ) N w r y umbr of ways to form sampl wth xactly rd chps Ths law s calld th hypr gomtrc dstrbuto. Exampl: Ur cotas rd chps, wht chps, ad blu chps. Ur II has rd, wht, ad 5 blu. Two chps ar draw at radom ad wthout rplacmt from ach ur. What s th probablty that all four chps ar th sam color? a S S robablty of All Sam Color R&R or W&W or B&B II I R W B R W 5 B

8 8 Exampl, How may throws of far dc ar dd to guarat that at last o 7 appars wth probablty of 0.8 or mor? robablty of throwg a sv quals /6. Th, at last o o tosss tosss solv for l 5 l Exampl: A dog food salsma slls hous to hous. From log xprc h has foud that 5% of th houss hav o dogs, 50% hav o dog, ad 5% hav two dogs. Corrspodgly, h has foud that h slls o bags, o small bag, or o larg bag, rspctvly, to ach such hous h vsts. Suppos h vsts four houss. What s th probablty that h wll sll ach th followg amouts of dog food? Solutos: a) o bags of dog food b) xactly two small bags ad o larg bag c) at last o small ad o larg bag a) No bags four calls corrspods to four vsts wth o dogs. o bags of 0 0 dog food ,0,0 four calls ( ) ( ) ( ) ( ) 09

9 9 xactly small & o l arg calls,, b) ( ) ( ) ( ) c) Cosdr, "at last larg & o small" mas xactly larg & small, or xactly larg & small, or xactly larg & small, or xactly larg & small, or xactly larg & small. Th,,, + 0,, + 0,, 0.65 ( ) ( 0.5) ( 0.5)( 0.5) + ( 0.5)( 0.5) ( 0.5) 0 ( 0.5) ( 0.5) ( 0.5) + ( 0.5)( 0.5)( 0.5) 0 ( 0.5) ( 0.5) ( 0.5),,,, As a prfac to what w wat to do xt, w cosdr ow a vry mportat probablty law. Gaussa Law: f x ( x) πσ x m σ

10 0 Ara a πσ x m σ dx - No closd form soluto xsts. Hc, umrcal soluto rqurd. W would hav a ft umbr of tabls (for all possbl valus of m ad σ), so w ormalz. Lt, x M z dz dx a M z a µ z σ Ara π z z dz Ths ar ow tabulatd a appdx all txt boos as valus of z Φ z ( z) dz to π ad th valus rprst th ara udr th ormalzd Gaussa Curv from Thr s also th Error Fucto, whch s tmatly rlatd to th Normalz Gaussa fucto; rf ( ) Φ( z) z Ara udr curv from org to z rf ( z) s symtrcal f(x) B a b X

11 Ara B π b µ z σ a µ z σ z dz Also, w could wrt b µ rf µ rf a σ σ Not: Φ( z ) Φ( + z) or rf ( z) rf ( + z) Lt us ow rvw I applyg Broull Trals w vry oft wat to dtrm th probablty of succsss bg btw two valus... { } p q For our lttl samplg problm: A ordr of 000 parts s rcvd. Th probablty of a dfctv tm s 0. W wat to fd th robablty that th total umbr of dfctv arts dos ot xcd ,000 ( ) p a ( ) p a part s dfctv q p 0, 00 W wat th probablty that th umbr of dfctv parts dos ot xcd 00. So, w must valuat, 00 0,000 { 0 00} ( 0.) ( 0.9) 0 0,000 Whch s qut dffcult, although wth computr programs such as MALE, Mathmatca ad MATLAB, w ca valuat ths. Howvr, w wat to st up th drvato for th Gassa Dstrbuto Law. So, w hav to ma som smplfyg approxmatos. O s volvd wth Error fucto, or Stadard Normal Fucto.

12 Now, f w assum; ) s vry larg (.. 0,000, tc.) ) pq>>> (900>>>) ) p s of ordr pq, It ca b show (Fllr - "Thory of robablty ad Applcato") that p q πpq ( p) pq Ths s ow as th DMovr-Lalac Thorm ad rsmbls th Gaussa Equato from abov. xσ ( x µ ) σ W s that th approxmato s th Gaussa Curv ctrd at p (most lly valu) Sdlght: Not, p <<< pq (ot our cas). Th, p q πpq (uform dstrbuto) Wll, usg thos rsults, w hav, { } p q πpq ( p) pq