2. Independence and Bernoulli Trials

Transcription

1 . Ideedece ad Beroull Trals Ideedece: Evets ad B are deedet f B B. - It s easy to show that, B deedet mles, B;, B are all deedet ars. For examle, ad so that B or B B B B B φ,.e., ad B are deedet evets., B; B B B B + B B + B B B B B B,

2 s a alcato, let ad q rereset the evets ad "the rme dvdes the umber N" "the rme q dvdes the umber N". q The from -4 { }, { q} q lso { q} {" q dvdes N"} { } { q} q - Hece t follows that ad q are deedet evets!

3 If 0, the sce the evet B always, we have B 0 B 0, ad - s always satsfed. Thus the evet of zero robablty s deedet of every other evet! Ideedet evets obvously caot be mutually exclusve, sce > 0, B > 0 ad, B deedet mles B > 0. Thus f ad B are deedet, the evet B caot be the ull set. More geerally, a famly of evets { } are sad to be deedet, f for every fte sub collecto,,,, we have. -3 3

4 4 Let a uo of deedet evets. The by De-Morga s law ad usg ther deedece Thus for ay as -4 a useful result., , -6

5 Examle.: Three swtches coected arallel oerate deedetly. Each swtch remas closed wth robablty. a Fd the robablty of recevg a ut sgal at the outut. b Fd the robablty that swtch S s oe gve that a ut sgal s receved at the outut. s Iut s s 3 Outut Fg.. Soluto: a. Let Swtch S s closed. The 3. Sce swtches oerate deedetly, we have, j j ;

6 Let R ut sgal s receved at the outut. For the evet R to occur ether swtch or swtch or swtch 3 must rema closed,.e., R Usg ,. 3 3 R R R + R. R, R We ca also derve -8 a dfferet maer. Sce ay evet ad ts comlmet form a trval artto, we ca always wrte But ad ad usg these -9 we obta R whch agrees wth -8. 3,

7 7 Note that the evets,, 3 do ot form a artto, sce they are ot mutually exclusve. Obvously ay two or all three swtches ca be closed or oe smultaeously. Moreover, b. We eed From Bayes theorem Because of the symmetry of the swtches, we also have R R R R R R R -

8 Reeated Trals Cosder two deedet exermets wth assocated robablty models Ω, F, ad Ω, F,. Let ξ Ω, η Ω rereset elemetary evets. jot erformace of the two exermets roduces a elemetary evets ω ξ, η. How to characterze a arorate robablty to ths combed evet? Towards ths, cosder the Cartesa roduct sace Ω Ω Ω geerated from Ω ad Ω such that f ξ Ω ad η Ω, the every ω Ω s a ordered ar of the form ω ξ, η. To arrve at a robablty model we eed to defe the combed tro Ω, F,. 8

9 Suose F ad B F. The B s the set of all ars ξ, η, where ξ ad η B. y such subset of Ω aears to be a legtmate evet for the combed exermet. Let F deote the feld comosed of all such subsets B together wth ther uos ad comlmets. I ths combed exermet, the robabltes of the evets Ω ad Ω B are such that Ω, Ω B B. Moreover, the evets Ω ad Ω B are deedet for ay F ad B F. Sce B we coclude usg - that - Ω Ω B, -3 9

10 B Ω Ω B B -4 for all F ad B F. The assgmet -4 exteds to a uque robablty measure o the sets F ad defes the combed tro Ω, F,. Geeralzato: Gve exermets Ω,, Ω, Ω, ad ther assocated F ad,, let Ω Ω Ω Ω -5 rereset ther Cartesa roduct whose elemetary evets are the ordered -tules ξ,, ξ, ξ, where ξ Ω. Evets ths combed sace are of the form -6 where, ad ther uos a tersectos. F 0

11 If all these exermets are deedet, ad s the robablty of the evet F the as before Examle.: evet has robablty of occurrg a sgle tral. Fd the robablty that occurs exactly tmes, trals. Soluto: Let Ω, F, be the robablty model for a sgle tral. The outcome of exermets s a -tule where every ξ Ω ad Ω as Ω Ω Ω The evet occurs at tral #, f ξ. Suose occurs exactly tmes ω.. -7 { ξ ξ,, ξ }, ω -8, Ω 0

12 The of the ξ belog to, say ξ, ξ,,, ad the ξ remag are cotaed ts comlmet. Usg -7, the robablty of occurrece of such a ω s gve by 0 ω { ξ, ξ,, ξ,, ξ } { ξ } { ξ } { ξ } { ξ } However the occurreces of ca occur ay artcular locato sde ω. Let ω, ω,, ω N rereset all such evets whch occurs exactly tmes. The But, all these s are mutually exclusve, ad equrobable. q. -9 " occurs exactly tmes trals" ω ω ωn. -0 ω

13 Thus " occurs exactly tmes trals" N ω N ω N q, - 0 where we have used -9. Recall that, startg wth ossble choces, the frst object ca be chose dfferet ways, ad for every such choce the secod oe ways, ad the th oe + ways, ad ths gves the total choces for objects out of to be + But, ths cludes the! choces amog the objects that are dstgushable for detcal objects. s a result 0. N +!!!! - 3

14 reresets the umber of combatos, or choces of detcal objects tae at a tme. Usg - -, we get " occurs a formula, due to Beroull. q exactly tmes trals", 0,,,,, -3 Ideedet reeated exermets of ths ature, where the outcome s ether a success or a falure are characterzed as Beroull trals, ad the robablty of successes trals s gve by -3, where reresets the robablty of success ay oe tral. 4

15 Examle.3: Toss a co tmes. Obta the robablty of gettg heads trals? Soluto: We may detfy head wth success ad let H. I that case -3 gves the desred robablty. Examle.4: Cosder rollg a far de eght tmes. Fd the robablty that ether 3 or 4 shows u fve tmes? Soluto: I ths case we ca detfy Thus { f 3 } { f }. " success" { ether 3 or 4 } 4 f3 + f ad the desred robablty s gve by -3 wth, ad /3. Notce that ths s smlar to a based co roblem. 3, 8 5 5

16 Beroull tral: cossts of reeated deedet ad detcal exermets each of whch has oly two outcomes or wth, ad q. The robablty of exactly occurreces of such trals s gve by -3. Let X " exactly occurreces trals". -4 Sce the umber of occurreces of trals must be a teger 0,,,,, ether X0 or X or X or or X must occur such a exermet. Thus X X X But X, are mutually exclusve. Thus X j 6

17 0 X X X q. 0 0 X From the relato a -6 equals + q, ad t agrees wth -5. For a gve ad what s the most lely value of? From Fg.., the most robable value of s that umber whch maxmzes -3. To obta ths value, cosder the rato -6 + b a b, -7 0, /. Fg.. 7

18 +! q +!!!!! q + q. -8 Thus, f + Thus as a fucto of creases utl or f t s a teger, or the largest teger max less tha ad -9 reresets the most lely umber of successes or heads trals. + Examle.5: I a Beroull exermet wth trals, fd the robablty that the umber of occurreces of s betwee ad., 8

19 Soluto: Wth X, 0,,,,, as defed -4, clearly they are mutually exclusve evets. Thus " Occurreces of X Examle.6: Suose 5,000 comoets are ordered. The robablty that a art s defectve equals 0.. What s the robablty that the total umber of defectve arts does ot exceed 400? Soluto: Let Y " X arts s betwee ad X X. q -30 " + are defectve amog 5,000 comoets". 9

20 Usg -30, the desred robablty s gve by Y 0 Y Y Y