CS 109 Lecture 12 April 22th, 2016

Transcription

1 CS 09 Lecture Aprl th 06

2 Four rototypcal Traectores Today:. Mult varable RVs. Expectato wth multple RVs 3. Idepedece wth multple RVs

3 Four rototypcal Traectores Revew

4 Dscrete Jot Mass Fucto For two dscrete radom varables ad the Jot robablty Mass Fucto s: p Margal dstrbutos: a b a b p a a p a y Example: value of de D value of de D y p b b p x b x 6 6 p y y y 36 6

5 robablty Table States all possble outcomes wth several dscrete varables Ofte s ot parametrc If #varables s > you ca have a probablty table but you ca t draw t o a slde All values of A a All values of B b A a B b Remember meas ad Every outcome falls to a bucet

6 robablty Table

7 Jotly Cotuous Radom varables ad are Jotly Cotuous f there exsts DF f x y defed over < x y < such that: a < a b < b f 900 a a b b x y dy dx y 0 x 900

8 Jotly Cotuous a < a b < b f a a b b x y dy dx Ca calculate probabltes f x y y b b a a x

9 Darts! y Ca calculate margal probabltes -xel Margal x -xel Margal N N

10 Four rototypcal Traectores Trasfer Learg

11 Four rototypcal Traectores Way Bac

12 ermutatos How may ways are there to order dstct obects?!

13 Multomal How may ways are there to order obects such that: are the same dstgushable are the same dstgushable r are the same dstgushable?!!!... r!... r Called the multomal because of somethg from Algebra

14 Bomal How may ways are there to mae a uordered selecto of r obects from obects? How may ways are there to order obects such that: r are the same dstgushable r are the same dstgushable? r!! r! r Called the Bomal Mult -> B

15 Bomal Dstrbuto Cosder depedet trals of Berp rad. var. s umber of successes trals s a Bomal Radom Varable: ~ B p robablty of exactly successes Bomal # ways of orderg the successes p p p 0... robablty of each orderg of successes s equal mutually exclusve

16 Four rototypcal Traectores Ed Revew

17 Welcome Bac the Multomal Multomal dstrbuto depedet trals of expermet performed Each tral results oe of m outcomes wth respectve probabltes: p p p m where umber of trals wth outcome m p c c c c... m cm p p c c... cm where Jot dstrbuto m c ad Multomal # ways of orderg the successes c c... c c! c!! c m m!... p c m m robabltes of each orderg are equal ad mutually exclusve

18 6-sded de s rolled 7 tmes Roll results: oe two 0 three four 0 fve 3 sx Ths s geeralzato of Bomal dstrbuto Bomal: each tral had possble outcomes Multomal: each tral has m possble outcomes !!0!!0!3! 7! Hello De Rolls My Old Freds

19 robablstc Text Aalyss Igorg order of words what s probablty of ay gve word you wrte Eglsh? word the > word trasatlatc word Staford > word Cal robablty of each word s ust multomal dstrbuto What about probablty of those same words someoe else s wrtg? word probablty wrter you > word probablty wrter o-cs09 studet After estmatg word wrter from ow wrtgs use Bayes Theorem to determe wrter word for ew wrtgs!

20 Text s a Multomal Example documet: ay for Vagra wth a credt-card. Vagra s great. So are credt-cards. Rs free Vagra. Clc for free. 8 Vagra Free Rs Credt-card: For robablty of seeg ths documet spam spam It s a Multomal!!!!...! p vagrap free...p for The probablty of a word spam emal beg vagra

21 Old ad New Aalyss Authorshp of Federalst apers 85 essays advocatg ratfcato of US costtuto Wrtte uder pseudoym ublus o Really Alexader Hamlto James Madso ad Joh Jay Who wrote whch essays? o Aalyzed probablty of words each essay versus word dstrbutos from ow wrtgs of three authors Flterg Spam word Vagra wrter you << word Vagra wrter spammer

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23 Four rototypcal Traectores Expectato wth Multple Varables?

24 Jot Expectato E[] x xpx Expectato over a ot s t cely defed because t s ot clear how to compose the multple varables: Add them? Multply them? Lemma: For a fucto g we ca calculate the expectato of that fucto: E[g ] xy gx ypx y By the way ths also holds for sgle radom varables: E[g] x gxpx

25 Expected Values of Sums E[ ] E[] E[] Geeralzed: E E[ ] Holds regardless of depedecy betwee s

26 Septcal Chrs Wats a roof! Let g [ ] E[ ]E[g ] gx ypx y xy [x y]px y xy What a useful lemma By the defto of gxy Brea that sum to parts! Chage the sum of xy to separate sums That s the defto of margal probablty That s the defto of expectato xpx y ypx y xy xy x px y y px y x y y x xpx ypy x y E[]E[ ]

27 Four rototypcal Traectores Idepedece ad Radom Varables

28 Idepedet Dscrete Varables Two dscrete radom varables ad are called depedet f: Itutvely: owg the value of tells us othg about the dstrbuto of ad vce versa If two varables are ot depedet they are called depedet Smlar coceptually to depedet evets but we are dealg wth multple varables p x y p x p y for all x Keep your evets ad varables dstct ad clear! y

29 Flp co wth probablty p of heads Flp co a total of m tmes Let umber of heads frst flps Let umber of heads ext m flps ad are depedet Let Z umber of total heads m flps Are ad Z depedet? o What f you are told Z 0? y m y x x p p y m p p x y x y x Co Flps

30 Let N # of requests to web server/day Suppose N ~ o Each request comes from a huma probablty p or from a bot probablty p depedetly # requests from humas/day N ~ BN p # requests from bots/day N ~ BN - p Web Server Requests robablty of huma requests ad bot requests robablty of umber of requests a day was robablty of huma requests ad bot requests we got requests

31 Let N # of requests to web server/day Suppose N ~ o Each request comes from a huma probablty p or from a bot probablty p depedetly # requests from humas/day N ~ BN p # requests from bots/day N ~ BN - p Note: Web Server Requests 0 ou got huma requests ad bot requests ou dd ot get requests

32 Let N # of requests to web server/day Suppose N ~ o Each request comes from a huma probablty p or from a bot probablty p depedetly # requests from humas/day N ~ BN p # requests from bots/day N ~ BN - p Web Server Requests

33 Let N # of requests to web server/day Suppose N ~ o Each request comes from a huma probablty p or from a bot probablty p depedetly # requests from humas/day N ~ BN p # requests from bots/day N ~ BN - p Web Server Requests! e p p! e p p

34 Let N # of requests to web server/day Suppose N ~ o Each request comes from a huma probablty p or from a bot probablty p depedetly # requests from humas/day N ~ BN p # requests from bots/day N ~ BN - p Where ~ op ad ~ o p ad are depedet! Web Server Requests!!!! e p p!! p p e!! p p p p e e

35 Idepedet Cotuous Varables Two cotuous radom varables ad are called depedet f: a b a b for ay a b Equvaletly: F f a b a b F f a F a f b b for all a b for all a b More geerally ot desty factors separately: f x y h x g y where < x y <

36 op Quz ust ddg Cosder ot desty fucto of ad : 3x y f x y 6e e for 0 < x y < Cosder ot desty fucto of ad : x y 4xy for 0 < x y Are ad depedet? Now add costrat that: 0 < x y < Are ad depedet? No! o Are ad depedet? es! 3x y Let h x 3e ad g y e so f Let f < es! x y h x g y h x x ad g y y so f x y h x g y Caot capture costrat o x y factorzato!

37 Datg at Staford Two people set up a meetg for pm 30 Each arrves depedetly at tme uformly dstrbuted betwee pm ad :30pm # m. past pm perso arrves ~ U0 30 # m. past pm perso arrves ~ U0 30 What s frst to arrve wats > 0 m. for other? 0 < 0 < 0 < y 0 y0 x dxdy < 30 x 0< y 30 y 0 y 0 30 y 30 0 f x y0 x 0 dx dy 30 y dxdy y 0 f x 0< y x y x f dy by symmetry y dxdy y y 0 dy

38 radom varables are called depedet f: Aalogously for cotuous radom varables: a a a a a a a... subsets of for all... x x x x x x x... subsets of for all... Idepedece of Multple Varables

39 Idepedece s Symmetrc If radom varables ad depedet the depedet of ad depedet of Duh!? Duh deed... Let... be a sequece of depedet ad detcally dstrbuted I.I.D. cotuous radom vars Say > for all... -.e. max... o Call a record value Let evet A dcate s record value o Is A depedet of A? o Is A depedet of A? o o Easer to aswer: es! By symmetry A / ad A / o A A // A A

40 Four rototypcal Traectores Earth Day

41 Choosg a Radom Subset From set of elemets choose a subset of sze possbltes are equally lely Oly have radom whch smulates ~ U0 such that all Brute force: Geerate a orderg of all subsets of sze Radomly pc oe dvde 0 to tervals Expesve wth regard to tme ad space Bad tmes!

42 Happly Choosg a Radom Subset I[] Good tmes: t dcatordouble p { f radom < p retur ; else retur 0; } // array I[] dexed from to subset rsubset set of sze { subset_sze 0; I[] dcatordouble/; for ; < ; { subset_sze I[]; I[] dcator subset_sze/ ; } retur subset cotag elemet[] ff I[] ; } I[ ] ad I[ ] I[]... I[ ] where < <

43 Radom Subsets the Happy Way roof Iducto o :.e. why ths algorthm wors Base Case: Set S {a} rsubset returs {a} wth p Iductve Hypoth. IH: for x c Gve set S S x ad x rsubset returs ay subset S of S where S wth p Iductve Case : where c S x I[] o Elem subset choose elems from remag o By IH: rsubset returs subset S of sze wth p o I[] subset S Iductve Case : where c S x I[] 0 o Elem ot subset choose elems from remag o By IH: rsubset returs subset S of sze wth p o I[] 0 subset S x

44 Sum of Idepedet Bomal RVs Let ad be depedet radom varables ~ B p ad ~ B p ~ B p Ituto: has trals ad has trals o Each tral has same success probablty p Defe Z to be trals each wth success prob. p Z ~ B p ad also Z More geerally: ~ B p for N N ~ B N p

45 Sum of Idepedet osso RVs Let ad be depedet radom varables ~ o ad ~ o ~ o roof: ust for referece Rewrte as where 0 Notg Bomal theorem: so ~ o 0 0 e e e e 0 0 0!!!!!!!! e! 0!!!