Let. Then. k n. And. Φ npq. npq. ε 2. Φ npq npq. npq. = ε. k will be very close to p. If n is large enough, the ratio n

Transcription

1 Let The m ( ) ( + ) where > very smll { } { ( ) ( + ) } Ad + + { } Φ Φ Φ Φ Φ Let, the Φ( ) lim This is lled thelw of lrge umbers If is lrge eough, the rtio will be very lose to.

2 Exmle -Tossig oi times. We wt to test the firess of oi. We defie "fir" to be tht we should get heds betwee 5% d 55% of the time. How my times should we toss to be.7% sure it is fir. Observe tht umber of heds is betwee.5 d.55 Hee, { } Assume fir oi ( ) Φ Φ

3 .5 ) )( ()( defie fir s heds omig u betwee d 5% of time { } ( ) 5. x x 3.6 d Φ Φ

4 Exmle The robbility of sseger hoosig irlie A d B is. Suose tht both irlies re ometig for the sme ool of ssegers. Airlie A sells tiets to everyoe who reuests oe, d the ity of its le is 3. Determie the robbility tht irlie A overboos., 3 Aroximtig this robbility, ideedet trils [ suess 3].5658 (Ext!) x 3 e dx Φ 3 Φ π 3 Φ Φ Φ( ) Φ( 3).87.3 ot so good estimte!!

5 But ( > 3) ( ) Φ Φ. + Φ (.) + Φ( ).8 +. Hee, (.3+.). 6 muh loser to tul vlue.

6 Exmle A erti tye of seed hs robbility of.8 of germitig. I ge of seeds, wht is the robbility tht t lest 75% will germite?.8, Arximtig 8 Φ 758, 8, 6 [ suess >.75] (.8) (.).5 (Ext!) π e ( 5) Φ(.5) But, we lso hve: 75 x [ > 75] [ 7] 8 dz Φ 6 7 Φ Φ + Φ.67.3 e πσ 758 Φ Φ 6 x) ( σ dx Φ 3 + Φ The [ > 75] (.3 +.8).3 whih is very lose to the ext vlue. ( ) 5

7 Let ; i suh fshio tht verge vlue. ( ) [ ] ( ) ( + ) suess is trils ( )! Multily to d bottom by + +!..! ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + [ ] Note : ( ) ( ) Also ( ) ( )! ( ) d from the defiitio of the turl umber, e, e Lim beuse z Lim lim ( z) z ( ) Lim ( ) ostt ( )( ) s ( ) µ [ ] e µ +, whe is ostt.

8 µ, M ( suess i trils) lim ( ) oisso Theorem e µ! "If the robbility of suess i sigle tril,, rohes while the umber of trils,, beomes ifiite i suh mer tht the me, µ, remis fixed, the the biomil distributio will roh the oisso distributio with me µ." We iterret this s: where t T { suessesi time itervlt } where λ T me rte of { } suess λt e e d ( λt )! ( ) ( λt )! e λt! me time betwee T λ suesses

9 Exmle A hromosome muttio believed to be lied with olor blidess is ow to our, o verge, oe i, births. If, bbies re bor this yer i erti ity, wht is robbility tht t lest oe will develo olor blidess? Wht is ext robbility model tht lies here. Solutio; t lest olorblid hild Aroximte vlue: but x e e x x x.8667 ext vlue λ! t (.) (.) x µ ( λt ) e ( µ ) x! µ λt e, hee x x! x.8666 x x x x -.353

10 Exmle ) Suose tht % of ll items i suermret re umred. A ustomer buys items d roeeds to he out through the exress le. Estimte the robbility tht the ustomer will be delyed beuse oe or more of the items reuire rie he. t µ λ t. (.). x x Aroximt e vlue : x [,,. ] (.) (.) [ x or more] ( x ). e ( M ).5 x.56 tul! ) Astroomers estimte tht s my s billio strs i the Mily Wy glxy my be eirled by lets. Let deote the robbility tht y suh solr system otis itelliget life. How smll be d still give 5-5 he tht there is itelliget life i t lest oe other solr system i our glxy? x x [,,.. ] [ ] ( x ) e M.5! ( x ) l e l.5 µ ( ) ( ) +.635x, robbility e x 6.x y oe hs life ( x )

11 Negtive Biomil The first suess will our o the th tril [ N ] ( ) geometri distributio The tril for the th suess our", W Let N i be umber of trils betwee (i-) th d i th suess. N + N + + N ll the N i umber of trils for re ideedet th suess w w ( W ) ( ) w robbility distributi o of the # of Negtive Biomil distributio w, + trils w, it tes for...suesses

12 Exmle If left-tur le of trffi's light hs ity of three rs, wht is the robbility tht the le will ot be suffiietly lrge to hold ll the drivers who wt to tur left i strig of six rs delyed by red sigl? Where the verge - 3% of ll rs wt to tur left. The robbility we wt is tht of the evet A { demd for servie (left tur) will begreter th 3} th {# of trils to suessis or more} Let w - umber of trils to th suess. [ 3 w 6] [ A] (.3) (.3) (.3) + (.3) (.3) + (.3) (.3).7 6 w w w

13 Exmle: Gmbler's Rui Two lyers, sy A d B, ly series of ideedet gmes, with A's robbility of wiig o eh tril beig d B's robbility of wiig o eh tirl beig -. The lyers eh wger oe uit o eh ideet tril. They otiue to ly util oe or the other's fortue is lost. If A begis with uits d B begis with - uits, fid the robbility tht A wis B's etire fortue? - + Let B Wis Figure.: A Shemti Digrm of the Gmbler's Rui roblem robbility Aeveully wisstrtigwith uits. + + (.) A Wis [ E] [ E wi o ext tril ] + [ E lose o ext tril ] + +. r + r, r or r. (.) 3. ( ) [ r + r ] (.)

14 (5.) + r r Solvig for r, we obti ( ) ( ) ( ) (6.), d, ± ± + ± ± ± r r r r The geerl solutio is ( ) (7.)., + ( ) ( ) + + Thus our solutio ( ) ( ) ( ) ( ) ( ).,,

15 For fir gme, r r. A th order reurree eutio must hve ideedet solutios.. Ad so for we obti +,,. This imlies tht (.) + + hee,,. (.) For fir gme, the robbility of wiig begiig with uits is diretly roortiol to.