CS 109 Lecture 8 April 13th, 2016

Transcription

1 CS 109 Lecture 8 Aprl 13th, 2016

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3 Storg Data o DNA We wat to kow the probablty of 100 base pars beg corrupted? Probablty of each base par beg corrupted s base-pars X umber of bts corrupted. X ~ B(4, 0.1) I real etworks, sed large bt strgs (legth 10 4 ) Probablty of bt corrupto s very small p 10-6 X ~ B(10 4, 10-6 ) s uweldy to compute Extreme ad p values arse may cases # bt errors fle wrtte to dsk (# of typos a book) # vstors to a popular webste # of servers crashes a day gat data ceter

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5 Facebook Lkes Let s try ad calculate the probablty of dfferet u N s the umber of people who see your post Maxmum Lkelhood estmate Set N to be value that maxmzes: N m N m X P ) (

6 Beroull vs Bomal Beroull s a type of RV that ca take o two values, 1 (for success) wth probablty p ad 0 (for falure) wth probablty (1- p) Bomal s the sum of Beroulls

7 Storg Data o DNA All the moves, mages, emals ad other dgtal data from more tha 600 basc smartphoes (10,000 ggabytes) ca be stored the fat pk smear of DNA at the ed of ths test tube.

8 Whther the Bomal Recall example of sedg bt strg over etwork 4 bts set over etwork where each bt had depedet probablty of corrupto p 0.1 X umber of bts corrupted. X ~ B(4, 0.1) I DNA (ad real etworks), sed large strgs (legth 10 4 ) Probablty of corrupto s very small p 10-6 X ~ B(10 4, 10-6 ) s uweldy to compute Extreme ad p values arse may cases # bt errors steam set over a etwork # vstors to a popular webste # of servers crashes a day gat data ceter # Facebook log requests that go to partcular server

9 Recall the Bomal dstrbuto Let p (equvaletly: p /) Whe s large, p s small, ad s moderate : Yeldg: p p X P ) (1 )!!(! ) ( X P ) / (1 ) / (1! 1 )!!(! ) ( 1) 1)...( ( + e ) / (1 1 1) 1)...( ( + 1 ) / 1 ( e e X P! 1! 1 ) ( Bomal the Lmt

10 X s a Posso Radom Varable: X ~ Po() X takes o values 0, 1, 2 ad, for a gve parameter > 0, has dstrbuto (PMF): Note Taylor seres: So:! ) ( e X P !... 2! 1!! 0 e 1!! ) ( e e e e X P Posso Radom Varable

11 Sedg Data o Network Redux Recall example of sedg bt strg over etwork P( X Sed bt strg of legth 10 4 Probablty of (depedet) bt corrupto p 10-6 X ~ Po( 10 4 * ) What s probablty that message arrves ucorrupted? 0) e! e Usg Y ~ B(10 4, 10-6 ): P( Y 0) 0.01 (0.01) 0! Caveat emptor: Bomal computed wth bult- fucto pytho software package, so some approxmatos may have occurred. Approxmatos are closer to you tha they may appear some software packages.

12 Smeo-Des Posso Smeo-Des Posso ( ) was a prolfc Frech mathematca Publshed hs frst paper at 18, became professor at 21, ad publshed over 300 papers hs lfe He reportedly sad Lfe s good for oly two thgs, dscoverg mathematcs ad teachg mathematcs. I m gog wth Frech Mart Freema

13 Posso s Bomal the Lmt Posso approxmates Bomal where s large, p s small, ad p s moderate Dfferet terpretatos of "moderate" > 20 ad p < 0.05 > 100 ad p < 0.1 Really, Posso s Bomal as à ad p à 0, where p

14 B(10,0.3) vs B(100,0.03) vs Po(3) 0.3 P(X k) B(10, 0.3) B(100, 0.03) Po(3) k

15 Teder (Cetral) Momets wth Posso Recall: Y ~ B(, p) E[Y] p Var(Y) p(1 p) X ~ Po() where p ( à ad p à 0) E[X] p Var(X) p(1 p) (1 0) Yes, expectato ad varace of Posso are same o It brgs a tear to my eye

16 A Real Lcese Plate See at Staford No, t s ot me but I kd of wsh t was.

17 It s Really All About Ras Cake Bake a cake usg may rass ad lots of batter Cake s eormous ( fact, ftely so ) Cut slces of moderate sze (w.r.t. # rass/slce) Probablty p that a partcular ras s a certa slce s very small (p 1/# cake slces) Let X umber of rass a certa cake slce total # rass X ~ Po(), where # cake slces

18 CS Bakg Ras Cake wth Code Hash tables strgs rass buckets cake slces Server crashes data ceter servers rass lst of crashed maches partcular slce of cake Facebook log requests (.e., web server requests) requests rass server recevg request cake slce

19 Defectve Chps Computer chps are produced p 0.1 that a chp s defectve Cosder a sample of 10 chps What s P(sample cotas 1 defectve chp)? Usg Y ~ B(10, 0.1): P( Y 1) 10 (0.1) 0 0 (1 0.1) (0.1) 1 1 (1 0.1) Usg X ~ Po( (0.1)(10) 1) P( X 1) e + e 0! 1! 2e

20 Effcetly Computg Posso Let X ~ Po() Wat to compute P(X ) for multple values of E.g., Computg Iteratve formulato: Compute P(X + 1) from P(X ) Use recurrece relato: 1! / 1)! /( ) ( 1) ( e e X P X P ) ( 1 1) ( X P X P + + e e X P 0! 0) ( 0 a X P a X P 0 ) ( ) (

21 Posso s all about the Meh Posso ca stll provde a good approxmato eve whe assumptos are mldly volated Posso Paradgm Ca apply Posso approxmato whe... Successes trals are ot etrely depedet o Example: # etres each bucket large hash table Probablty of Success each tral vares (slghtly) o o Small relatve chage a very small p Example: average # requests to web server/sec. may fluctuate slghtly due to load o etwork

22 Brthday Problem Redux What s the probablty that of m people, oe share the same brthday (regardless of year)? l( e trals, oe for each par of people (x, y), x y Let E x,y x ad y have same brthday (tral success) P(E x,y ) p 1/365 (ote: all E x,y ot depedet) m 1 m( m 1) X ~ Po() where m( m1)/730 ( m( m 1) / 730) m( m P( X 0) e e 0! m( m1)/ 730 Solve for smallest teger m, s.t.: e 0.5 m m 2 ( m1)/730 ) l(0.5) Same as before! 1)/730 m( m 1) 730l(0.5) m 23

23 Posso Process Cosder rare evets that occur over tme Earthquakes, radoactve decay, hts to web server, etc. Have tme terval for evets (1 year, 1 sec, whatever...) Evets arrve at rate: evets per terval of tme Splt tme terval to à sub-tervals Assume at most oe evet per sub-terval Evet occurreces sub-tervals are depedet Wth may sub-tervals, probablty of evet occurrg ay gve sub-terval s small N(t) # evets orgal tme terval ~ Po()

24 Web Server Load Cosder requests to a web server 1 secod I past, server load averages 2 hts/secod X # hts server receves a secod What s P(X 5)? Model Assume server caot ackowledge > 1 ht/msec. 1 sec 1000 msec. ( large ) P(ht server 1 msec) 2/1000 ( small p) X ~ Po( 2) P( X 5) e !

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26 Hurrcaes per Year sce Number of Hurrcaes Atlatc

27 Four Prototypcal Trajectores To the code!

28 Hstorcally ~ Posso(8.5) Predcted ad Actual Frequecy Utl 1966, thgs look pretty Posso Lambda Num Hurrcaes

29 Improbablty Drve What s the probablty of over 15 hurrcaes a seaso gve that the dstrbuto does t chage? Let X # hurrcaes a year. X ~ Po(8.5) Soluto: P (X >15) 1 P (X apple 15) 1 X P (X ) Ths s the pdf of a Posso. Your favorte programmg laguage has a fucto for t

30 Four Prototypcal Trajectores Twce sce 1966 there have bee years wth over 30 hurrcaes

31 Improbablty Drve What s the probablty of over 30 hurrcaes a seaso gve that the dstrbuto does t chage? Let X # hurrcaes a year. X ~ Po(8.5) Soluto: P (X >30) 1 P (X apple 30) 1 X30 0 P (X ) e 09 Ths s the pdf of a Posso. Your favorte programmg laguage has a fucto for t

32 The Dstrbuto has Chaged 8 Predcted ad Actual Frequecy Sce 1966, looks lke the dstrbuto has chaged Lambda 16.6? # Hurrcaes

33 What s Up?

34 Four Prototypcal Trajectores Pause

35 Dscrete Dstrbutos Do t have to memorze all of the followg dstrbutos. We wat you to get a sese of how radom varables work

36 Geometrc Radom Varable X s Geometrc Radom Varable: X ~ Geo(p) X s umber of depedet trals utl frst success p s probablty of success o each tral X takes o values 1, 2, 3,, wth probablty: P( X ) (1 p) 1 p E[X] 1/p Var(X) (1 p)/p 2 Examples: Flppg a co (P(heads) p) utl frst heads appears Ur wth N black ad M whte balls. Draw balls (wth replacemet, p N/(N + M)) utl draw frst black ball Geerate bts wth P(bt 1) p utl frst 1 geerated

37 Negatve Bomal Radom Varable X s Negatve Bomal RV: X ~ NegB(r, p) X s umber of depedet trals utl r successes p s probablty of success o each tral X takes o values r, r + 1, r + 2, wth probablty: E[X] r/p Var(X) r(1 p)/p 2 Note: Geo(p) ~ NegB(1, p) Examples: 1 r r P( X ) p (1 p), where r, r + 1,... r 1 # of co flps utl r-th heads appears # of strgs to hash to table utl bucket 1 has r etres

38 Hypergeometrc Radom Varable X s Hypergeometrc RV: X ~ HypG(, N, m) Ur wth N balls: (N m) black ad m whte Draw balls wthout replacemet X s umber of whte balls draw m N m P ( X ), where 0,1,..., N E[X] (m/n) Var(X) [m(n )(N m)]/[n 2 (N 1)] Let p m/n (probablty of drawg whte o 1 st draw) Note: HypG(, N, m) à B(, m/n) As N à ad m/n remas costat

39 Edagered Speces Determe N how may of some speces rema Radomly tag m of speces (e.g., wth whte pat) Allow amals to mx radomly (assumg o breedg) Later, radomly observe aother of the speces X umber of tagged amals observed group of X ~ HypG(, N, m) Maxmum Lkelhood estmate Set N to be value that maxmzes: P ( X ) m N N m