13. The Weak Law and the Strong Law of Large Numbers

Transcription

1 3. Th Wa Law ad th Strog Law of Larg Numbrs Jams Broull rovd th wa law of larg umbrs WLLN aroud 7 whch was ublshd osthumously 73 hs trats Ars Cojctad. osso gralzd Broull s thorm aroud, ad 66 Tchbychv dscovrd th mthod barg hs am. Latr o o of hs studts, Marov obsrvd that Tchbychv s rasog ca b usd to xtd Broull s thorm to ddt radom varabls as wll. I 99 th Frch mathmatca Eml Borl rovd a dr thorm ow as th strog law of larg umbrs that furthr gralzs Broull s thorm. I 96 Kolmogorov drvd codtos that wr cssary ad suffct for a st of mutually ddt radom varabls to oby th law of larg umbrs ṖILLAI

2 Lt X b ddt, dtcally dstrbutd Broull radom Varabls such that X, X, ad lt X X X rrst th umbr of succsss trals. Th th wa law du to Broull stats that [s Thorm 3-, ag, Txt]. 3-.., th rato total umbr of succsss to th total umbr of trals tds to robablty as crass. A strogr vrso of ths rsult du to Borl ad Catll stats that th abov rato / tds to ot oly robablty, but wth robablty. Ths s th strog law of larg umbrs SLLN.

3 What s th dffrc btw th wa law ad th strog law? Th strog law of larg umbrs stats that f s a suc of ostv umbrs covrgg to zro, th <. 3- From Borl-Catll lmma [s -69 Txt], wh 3- s satsfd th vts A ca occur oly for a ft umbr of dcs a ft suc, or uvaltly, th vts < occur ftly oft,.., th vt / covrgs to almost-surly. roof: To rov 3-, w rocd as follows. Sc 3

4 w hav ad hc whr By drct comutato < 3-3 X X E X E

5 E E Y 3 Y 3 j l j E Y 3 E YY Y Y 3 E Y 3 [ 3 ] E Y 3, 3- sc <, / < Substtutg 3- also 3-3 w obta j l j ca cocd wth j, or l, ad th scod varabl tas - valus E Y j j 3 Lt / so that th abov tgral rads ad hc 3/ 3 3 / x dx 3/ 3 9 <, 3-

6 thus rovg th strog law by xhbtg a suc of ostv / umbrs that covrgs to zro ad satsfs 3-. / W rtur bac to th sam usto: What s th dffrc btw th wa law ad th strog law?. Th wa law stats that for vry that s larg ough, th rato X / / s lly to b ar wth crta robablty that tds to as crass. Howvr, t dos ot say that / s boud to stay ar f th umbr of trals s crasd. Suos 3- s satsfd for a gv a crta umbr of trals. If addtoal trals ar coductd byod, th wa law dos ot guarat that th w / s boud to stay ar for such trals. I fact thr ca b vts for whch /, for som rgular mar. Th robablty for such a vt s th sum of a larg umbr of vry small robablts, ad th wa law s uabl to say aythg scfc about th covrgc of that sum. Howvr, th strog law stats through 3- that ot oly all such sums covrg, but th total umbr of all such vts 6

7 7 whr s fact ft! Ths mls that th robablty of th vts as crass bcoms ad rmas small, sc wth robablty oly ftly may volatos to th abov ualty tas lac as Itrstgly, f t ossbl to arrv at th sam cocluso usg a owrful boud ow as Brst s ualty that s basd o th WLLN. Brst s ualty : Not that ad for ay ths gvs Thus. /,.

8 Sc for ay ral x, Substtutg 3-7 to 3-6, w gt But s mmum for ad hc Smlarly. x x x.. / , / 3- / <

9 ad hc w obta Brst s ualty Brst s ualty s mor owrful tha Tchbyshv s ualty as t stats that th chacs for th rlatv frucy / xcdg ts robablty tds to zro xotally fast as. Chbyshv s ualty gvs th robablty of / to l btw ad for a scfc. W ca us Brst s ualty to stmat th robablty for / to l btw ad for all larg. Towards ths, lt so that / 3-9. y < c y To comut th robablty of th vt y, ot that ts m comlmt s gv by c c y y m m / 9

10 ad usg E. -6 Txt, y y Ths gvs m / c c /. m / m m m / y y as m m m / or,, for all m as m. Thus / s boud to stay ar for all larg ough, robablty, a cocluso alrady rachd by th SLLN. Dscusso: Lt.. Thus f w toss a far co, tms, from th wa law..

11 Thus o th avrag 39 out of such vts ach wth or mor trals wll satsfy th ualty or, t s ut ossbl. that o out of such vts may ot satsfy t. As a rsult f w cotu th co tossg xrmt for a addtoal mor trals, wth rrstg th total umbr of succsss u to th currt tral, for, t s ut ossbl that for fw such th abov ualty may b volatd. Ths s stll cosstt wth th wa law, but ot so oft says th strog law. Accordg to th strog law such volatos ca occur oly a ft umbr of tms ach wth a ft robablty a ft suc of trals, ad hc almost always th abov ualty wll b satsfd,.., th saml sac of / cocds wth that of as. Nxt w loo at a xrmt to cofrm th strog law: Examl: rd cards ad blac cards all dstct ar shuffld togthr to form a sgl dc, ad th slt to half. What s th robablty that ach half wll cota rd ad blac cards?

12 Soluto: From a dc of cards, cards ca b chos dffrt ways. To dtrm th umbr of favorabl draws of rd ad blac cards ach half, cosdr th uu draw cosstg of rd cards ad blac cards ach half. Amog thos rd cards, of thm ca b chos dffrt ways; smlarly for ach such draw thr ar ways of choosg blac cards.thus th total umbr of favorabl draws cotag rd ad blac cards ach half ar amog a total of draws. Ths gvs th dsrd robablty to b!!! For larg, usg Stglg s formula w gt.

13 [ π π [ π ] ] π For a full dc of cards, w hav 3, whch gvs. ad for a artal dc of cards that cotas rd ad blac cards, w hav ad.36. O summr aftroo, cards cotag rd ad blac cards wr gv to a yar old chld. Th chld slt that artal dc to two ual halvs ad th outcom was dclard a succss f ach half cotad xactly rd ad blac cards. Wth adult survso trms of shufflg th xrmt was ratd tms that vry sam aftroo. Th rsults ar tabulatd blow Tabl 3., ad th rlatv frucy vs th umbr of trals lot Fg 3. shows th covrgc of / to. 3

14 Tabl Numbr of succsss Ext Numbr of succsss Ext Numbr of succsss Ext Numbr of succsss Ext Numbr of succsss Ext

15 Th fgur blow shows rsults of a xrmt of trals..337 Fg 3.