Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs Itroucto to sttstcs... 7 Cotuous Dstrbutos... 9 Guss Dstrbuto (D)... Coutg evets to etere probbltes... Bol Coeffcets (Dstrbuto)... 3 Strlg s Appoto... 4 Guss Approto to bol strbuto for lrge... 5 Dervto of the Guss Dstrbuto... 6 Chpter : Probblty Sttstcs The essetl rguet sttstcl echc epes o probbltes. A prtculr cofgurto s fou wth cert probblty, we f propertes of sple by vergg proceure. Becuse the uber of olecules s so lrge ( ~ ) verges re er certtes, evtos fro the verge re eceegly sll (e.g. reltve error / N ~ ). I ths set of lectures I wll scuss: - Itroucto to sttstcs: verges str evtos - Cotuous strbutos, the orl or Guss strbuto - Coutg possbltes to rrve t probbltes - A relevt stt- ech eple: the bol strbuto Itroucto to sttstcs To trouce bsc ssues let us scuss sple eple: Throwg ce If you throw e, you wll get the result of,, 3, 4, 5, 6 ech throw. Suppose we throw the ce y tes (6) You get Possble results: =,,3, 4,5, 6 For fr ce, the rw of ech uber hs equl chce, the probbltes to throw 4 s /6 98 3 995 4 5 5 3 6 98 Totl 6 Chpter : Probblty Sttstcs 7
Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs We c efe the frctol occurrece s f 98 f = e. f =, f = etc. 6 6 6 tot I the lt of lrge uber of throws, ths uber wll pproch the probblty P of 6 Hece l f P = tot 6 We woul epect totp But the ctul ubers woul fluctute rou the epecte Averge: If the possble outcoes for vul eperet re, the uber of evets s the b = = = f tot = tot l f = P = tot A For ce: verge = ( + + 3+ 4+ 5+ 6) = = 3 6 6 Note tht the verge vlue y ot be possble result! = = ll ottos of verge Vrce: We re lso tereste the spre rou the e.. let us efe Vrce: = tot ( ) l = f P tot = P ( ) = P + P P P = + Use P = = P + = = P = = tot P = = = = tot tot tot Chpter : Probblty Sttstcs 8
Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs Ths verge of fro us the verge of squre s lwys greter th. Ths s esly see P, the se vlue P ( ). The spre s oly f every eperet yels You c esly verfy (for the eple gve) tht both wys of clcultg, ( ) P Str Devto:, yels the se result. Here we hve prove tht they re lwys the se. = The bove results cosere screte outcoes of cert eperet, =,,3..., but the lyss c be geerlze to cotuous strbutos. Cotuous Dstrbutos Let s coser cotuous strbuto p( ). Ths ght represet for eple ss strbuto log D le. Mss esty of leves ρ = ss betwee + b ρ ( ) = M the totl ss ρ s the ss betwee b For our curret purposes t s ore coveet to orlze ( ) P( ) M ρ = = b Such tht P s the frcto of the totl ss lyg the tervl [,b] P( ) s esoless qutty s clle the strbuto fucto over the vrble, t s logous to P = P( ). Here serves s our vrble. = P logous to P = P logous to P Chpter : Probblty Sttstcs 9
Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs P = = = P Eple: ( ) = posto = = p p Guss Dstrbuto (D) A fous strbuto tht we wll ecouter ore ofte s the Guss or orl strbuto / G = Ce : the wth of the strbuto (wll be show to be ) C : orlzto costt / G C e = = Also C = = (see below for ervto) π π / = e π = g () g( ) g s o fucto = / = e = π π / e π π = = ( show below). = = = s cle before Useful Guss Itegrl forul (ws use bove) k α y 3 5... ( k ) π y e y = k =,,3... ( k + ) k α α I ths cse k =, α = Chpter : Probblty Sttstcs
Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs (Plese ote the tegrto rge. For eve tegrs (w.r.t. ) oe c tke twce the result for the full tegrto rge.) Let us evlute crefully: = π e / = π e / = π π = Nothg essetl chges by shftg the u the strbuto wy fro ( y ) k e ( 3 5... k ) y =. α y Coutg evets to etere probbltes k+ α k A bsc strtegy to etere probbltes s s follows # of evets of terest P = totl # of possble evets Here we ssue ech evet tself to be eqully lkely. Eg. Throw co or ce or rwg cr fro eck To llustrte I wll use eples usg pck of crs: 4 suts: clubs, os, herts, spes 3 crs:, 3, 4, 5, 6, 7, 8, 9,, J, Q, K, A 5 crs totl ) Drw sequece of 5 crs ( poker h) where the orer oes tter # of st cr possbltes 5 # of cr possbltes 5 3 r 5 : : Hece the uber of possbltes of poker h where the orer oes tter s 5! 5 5 5 49 48 = 47! ) Wht bout the uber of peruttos the 5 crs where the orer oes tter # of st cr possbltes 5 # of cr possbltes 4 3 r 3 4 th 5 th # of peruttos (fferet cobtos) = 5 4 3 = 5! 3) Fro the prevous results: rwg sequece of 5 crs, where orer oes ot tter π α Chpter : Probblty Sttstcs
Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs 5 5 5 49 48 5! 5! # of sequeces = = = 5 4 3 47! 5! 47!5! 5 Ths c lso be wrtte s or C (5,5) 5 choose 5 5 So geerl, f we re to choose objects fro N totl objects, the orer of the cobtos oes tter N! N! Nuber of cobtos = The ore portt cse for us: If the orer of the cobto oes ot tter the N N! Nuber of cobtos = =! ( N )! Soe ore vce eples ) How y cobtos of 3 Quees + o Quee re there? There s Qs, Qh, Qc, Q. 4 4! 4 4 possbltes choose 3, = = = 4 3 3!! So there re 4 wys to rw 3 quees out of 4 A the o quees? There re 48 other crs (f you ot the lst quee) we pck, so we 48 get - ow wht s the probblty to rw 3 quees o quees? 4 48 3 Prob = 5 5 = (# of rw 5) # of 3 Q's (# of other) b) wht bout y trple ( + z + y)? ote tht z=y s clue but =y =z s eclue 4 48 3 3 Prob = 5 5 - full house? ( + yy) Chpter : Probblty Sttstcs
Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs 4 4 3 3 Prob = 5 5 You c check these results o goo poker wk pge! Bol Coeffcets (Dstrbuto) ( + b) = ( + b)( + b)( + b)...( + b) ( + b) = b = The bol / choose coeffcets for so clle Pscl trgle + b + b + b = + b+ b 3 3 3 3 3 + b = + 3 b+ 3b + b 4 6 4 5 5 You c scer the ptter strtg fro the top row Rtolzto: = b C b + = Drw tes out of, the orer oes ot tter C = Specl cse = p, p, b= q= p ( p+ q) = = = p q = P = p q P = the probblty to rw p tes whe rwg tes totl Chpter : Probblty Sttstcs 3
Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs Applcto of the bol Strlg s Appoto = # of prtcles o p se ( V L ) N = # of prtcles o q se ( V R ) = Np whch hppes to lso hve the hghest probblty 5! quckly becoes very lrge uber. I stt- ech ght be! For lrge eough uber Strlg s pproto s ccurte where s teger (screte vlue) l! l for screte vlues l! l for cotuous vlues l! ( l ) = l + = l Dervg Strlg s Approto ( ) ( ) ( ) l! = l... = l + l + l... + l = l for screte = If we go to cotuous we c replce the su wth tegrl Chpter : Probblty Sttstcs 4
Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs ( l ) = l I sury l = l = Δ l = l = l = = = = l l+ l (f s bg, ths pproto works very well) l! l l! l Guss Approto to bol strbuto for lrge (ot so portt, teous. Wll show usg Mtlb). For p+ q= p, we h the bol strbuto P p q = It s frly esy to show (see et pge) tht for lrge N, ths pproches the Guss strbuto where = Np = Npq PN e = e π π ( ) / ( Np) / Ths strbuto becoes cresgly rrow or hghly peke s creses. By ths we e tht N = Npq Np = q p N For eple f boes p= q=, we hve = prtcles whe strbutg the prtcles over verge uber of prtcles the left bo : Str evto: Npq = = So we epect the uber of prtcles to be ± Chpter : Probblty Sttstcs 5
Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs To be precse, for Guss strbuto to f result to be wth the e ± s 66.6 %, whle there s 99.9 % probblty to f the result betwee the e ±3. The portt pot s tht =!!! We woul ke very sll errors ssug ectly prtcles ech bo. Fluctutos re of orer ~ Dervto of the Guss Dstrbuto fro the bol strbuto N Let us ssue Tylor seres epso of l P rou ts u l PN = l PN + l PN ( ) l P N ( ).. + + N N N! N sce PN = p q = p q! ( N )! l P N () = l N! -l! l( N )! + l p + ( N )lq We kow fro Strlg s l! = l l! = l ( ) l ( N )! = N l N! = N So the st ervtves re l PN = l + l ( N ) + l p l q= t the u Note: ( l N!) = l P N N + = + = = N N N N At the u of the strbuto the frst ervtve goes to l P () = l + l N N + l p lq = N p l = q N p = q Np p = q Chpter : Probblty Sttstcs 6
Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs Np ( p q) = + = = = Np So the u s fou to be = Np. Ths s the epecte verge. Lookg t the seco ervtve N Usg = = Np = = Np N Np Np p Npq ( ) ( ) P N () = = Npq Gog bck to the tylor epso P l PN = l PN + l PN ( ) l P N ( )... + + l P N = l P N ( Npq ) +... P N e l P N () e ( ) / Npq ( ) /Npq PN = Ce where C = (orlzto costt) Npqπ Averge = = Np, vrce = = Npq Note e: Ths proof (wely quote tet books) s pretty b. You coul show ths wy (gog to seco orer oly) tht y strbuto wth u s Guss strbuto, whch s osese. Oe relly hs to show tht the hgher ervtves re eglgble. We t. I the coputer lb, we wll ke the coprso o coputer, you wll see tht ths pproto s ecellet. As so ofte the result s correct, the correct ervto s lckg. I t f t y otes. Perhps, f you uerst why ths proof s b, ths tself s useful thg to ler! Chpter : Probblty Sttstcs 7