Discrete Mathematics and Probability Theory Fall 2014 Anant Sahai Homework 11. This homework is due November 17, 2014, at 12:00 noon.

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1 EECS 70 Discrt Mathmatics ad Probability Thory Fall 2014 Aat Sahai Homwork 11 This homwork is du Novmbr 17, 2014, at 12:00 oo. 1. Sctio Rollcall! I your slf-gradig for this qustio, giv yourslf a 10, ad writ dow what you wrot for parts (a) ad (b) blow as a commt. Put th aswrs i your writt homwork as wll. (a) What discussio did you attd o Moday last wk? If you did ot attd sctio o that day, plas tll us why. (b) What discussio did you attd o Wdsday last wk? If you did ot attd sctio o that day, plas tll us why. 2. Biasd Cois, Birthday Paradox, ad Stirlig s Approximatio Lab Up util this poit, vrythig that you hav do i th last thr virtual labs is somthig that you could v aturally discovrd yourslf as somthig worth tryig. Th data is spakig dirctly to th xprimtalist i you. Howvr, discovrig a actual formula for th shap of this cliff-fac is somthig that actually rquirs a thortical ivstigatio that is rlatd to coutig, Fourir Trasforms, ad Powr Sris. Gussig its xact shap is ot somthig that coms vry aturally o xprimtalist ituitio alo. I this wk s lab, w will simply provid you with th right curv ad cotiu from last wk s lab o biasd cois. Ulss spcifid othrwis, you ca assum th sam cofiguratios from last wk s lab. I othr words, th coi is biasd with P(had) = 0.7, th umbr of tosss ar (k = 10,100,1000,4000), rspctivly, ad th umbr of trials is m = Mak sur you rviw th lab solutio from Homwork 10 bfor movig o. I additio, w will also look at th Birthday Paradox ad Stirlig s Approximatio. Plas com back to th last thr parts of th lab wh you ar workig o Qustio 8. For ach part, studts who wat to ca choos to compltly rwrit th qustio. Basically, you ca com up with your ow formulatio of how to do a sris of xprimts that rsult i th sam discovris. Th, writ up th rsults icly usig plots as appropriat to show what you obsrvd. You ca also rwrit th tir lab to tak a diffrt path through as log as thy covy th ky isights aimd at i ach part. Plas dowload th IPytho startr cod from Piazza or th cours wbpag, ad aswr th followig qustios. (a) Plot d 1 2π x2 2 dx ovrlaid with th ormalizd cliff-fac shaps you had plottd i last wk s lab. This itgral is rlatd to somthig calld th Error Fuctio. What do you obsrv? This is th hart of th Ctral Limit Thorm as applid to coi tosss. Hit: Implmt th fuctio ormal, which taks a ral umbr x ad rturs 1 2π x2 2. Th, implmt th fuctio itgrat_ormal(d), which itgrats th abov fuctio from to d. I Pytho, you ca us scipy.itgrat.quad. EECS 70, Fall 2014, Homwork 11 1

2 (b) Now, sic you had ralizd arlir that th cliff-facs ad th histograms hav som atural rlatioship with ach othr, how would you aturally ovrlay a smooth plot of 1 2π x2 2 to th ormalizd histograms. What dos this ma? Hit: Thr s a paramtr for plt.hist() you lard i HW7 that you ca us to ormaliz th histogram. Also, us a bi siz of 0.2. (c) Aothr itrstig pattr that you had s i th prvious Virtual Labs was th xpotial drop i th frqucis of crtai rar vts. For a xpotial drop, th most itrstig thig is to udrstad th rat of th xpotial or th rlvat slop o th Log-Liar plot. For a coi with probability p of big hads, w ar itrstd i th frqucy by which tossig k such cois rsults i mor tha ak hads (whr a is a umbr largr tha p). W ar itrstd i p = 0.3,0.7 ad a = p+0.05, p+0.1. Tak m = 1000 ad plot th atural log of th frqucis ths dviatios agaist k (ragig from 10 to 200). Approximatly xtract th slops for all four of ths. Compar thm i a tabl agaist th prdictios of th followig formula (which w will driv latr i th cours). D(a p) = al a 1 a + (1 a)l p 1 p. This xprssio is calld th Kullback-Liblr divrgc ad is also calld th rlativ tropy. Fially, add D(a p)k to th plots (thr should b 4 of ths) you hav mad as straight lis for immdiat visual compariso. This straight li is calld a Chroff Boud o th probability i qustio. What do you obsrv? Hit: First, implmt th fuctio KL, which computs D(a p) usig th giv formula. To fit a li i Pytho, you ca us p.polyfit. Thr will probably b som 0 valus, which will mss up this fittig, so you ca rplac th zros with (d) Durig your first wk of Charm School (CS), you wat to fid fllow CS studts who hav th sam birthday. Lt s switch gars to a itrstig problm studid i Lctur Not 12: th Birthday Paradox. This itrstig phomo cocrs th probability of two popl i a group of m popl havig th sam birthdays. This probability is giv by whr P(A c ) = 1 P(A) = (365 m + 1) 365! 365 m = 1 (365 m)!365 m ( 365! (365 m)!365 m = 1 1 ) ( 1 2 ) (... 1 m 1 ) ad P(A) is th probability that o two popl hav th sam birthday. For m = 10, 23, 50, 60, radomly grat birthdays by uiformly pickig m umbrs btw 1 ad 365. Do this 1000 tims for ach valu of m. Rcord how may trials hav at last 2 sam birthdays. Plot this fractio vs. m usig a bar chart. What do you obsrv? Hit: First, implmt th fuctio has_duplicat, which rturs Tru if a list cotais ay rpatd lmt ad Fals othrwis. Th, implmt th fuctio g_birthday(m), which grats radom birthday for m popl. () W will ow calculat th probability of havig two popl with th sam birthday mpirically, ad plot th rsult agaist th xpctd probability, which is drivd i Not 12. EECS 70, Fall 2014, Homwork 11 2

3 Implmt th fuctio birthday_formula(m), which calculats th probability of at last two popl havig th sam birthday amog m popl. Plot th mpirical rsult (you ca assum 1000 trials) v.s. th aalytical rsult (birthday_formula(m)), for m = [1,100] popl. What do you obsrv about th two curvs? What happs at m = 23 ad m = 60? Is this cosistt with what w prviously kw about th Birthday Paradox? (f) Now approximat P(A) usig Stirlig s approximatio for! ad plot th approximatd P(A c ) = 1 P(A) as a fuctio of m. Stirlig s approximatio is giv by! ( ) 2π. Plot th aalytical rsult from th prvious part ad th approximatd rsult i th sam figur. What do you obsrv? Hit: Implmt th fuctio birthday_stirlig, which computs th probability that o two popl hav th sam birthday giv that thr ar m birthdays usig Stirlig s approximatio. That said, do t us Stirlig s approximatio dirctly! Simplify your xprssio aftr usig th approximatio as much as possibl bfor you implmt th birthday_stirlig fuctio, or th larg valus will blow up your computr. (g) Lastly, lt s com back to th problm of coutig th umbr of ways to throw m balls ito bis. Suppos th umbr of balls i ach bi is a ogativ itgr, implmt th fuctio prmutatio(m,), which grats all possibl prmutatios of throwig m balls ito bis. For xampl, prmutatio(2,3) should rtur [[0,0,2],[0,1,1],[0,2,0],[1,0,1],[1,1,0],[2,0,0]]. How would you chag your implmtatio if w ow rquir ach bi to cotai a positiv umbr of balls? Hit: Us rcursio. You should hav thr bas cass. (h) Qustio 8, part (a) (i) Qustio 8, part (h) (j) Qustio 8, part (j) Rmidr: Wh you fiish, do t forgt to covrt th otbook to pdf ad mrg it with your writt homwork. Plas also zip th ipyb fil ad submit it as hw11.zip. 3. Pickig CS Classs Th EECS (Elgat Etiqutt Charm School) dpartmt has d diffrt classs big offrd i Fall Ths iclud classs such as drssig tiqutt, diig tiqutt, ad social tiqutt, tc. Lt s assum that all th classs ar qually popular ad ach class has sstially ulimitd satig! Suppos that c studts ar rolld this smstr ad th rgistratio systm, ElBEARS (Elgat Bars), rquirs ach studt to choos a class s/h plas to attd. (a) What is th probability that a giv studt chooss th first class, drssig tiqutt? (b) What is th probability that a giv class is chos by o studt? (c) If thr ar d = 20 classs, what should c b i ordr for th probability to b at last o half that (at last) two studts roll i th sam class? EECS 70, Fall 2014, Homwork 11 3

4 4. Sock tiqutt I your scod wk of Charm School you lar that you should oly war matchig pair of socks. I ach pair, both socks must b of th sam color ad pattr. But all of thm ar i o big baskt ad ow you hav to tak a pair out. Lt s say you ow pairs of socks which ar all prfctly distiguishabl (o two pairs hav th sam color ad pattr). You ar ow radomly pickig o sock aftr th othr without lookig at which o you pick. (a) How may distict substs of k socks ar thr? (b) How may distict substs of k socks which do ot cotai a pair ar thr? (c) What is th probability of formig at last o pair wh pickig k socks out of th baskt? (d) Now, i a diffrt xprimt, suppos thr is xactly o sock of ach pair i th baskt (so thr ar socks i th baskt) ad w sampl (with rplacmt) k socks from th baskt. What is th probability that w pick th sam sock at last twic i th cours of th xprimt? 5. Druk ma Imagi that you hav a druk ma movig alog th horizotal axis (that strtchs from x = to x = + ). At tim t = 0, his positio o this axis is x = 0. At ach tim poit t = 1, t = 2, tc., th ma movs forward (that is, x(t +1) = x(t)+1) with probability 0.5, backward (that is, x(t +1) = x(t) 1) with probability 0.3, ad stays xactly whr h is (that is, x(t + 1) = x(t)) with probability 0.2. (a) What ar all his possibl positios at tim t, t 0? (b) Calculat th probability of ach possibl positio at t = 1. (c) Calculat th probability of ach possibl positio at t = 2. (d) Calculat th probability of ach possibl positio at t = 3. () If you kow th probability of ach positio at tim t, how will you fid th probabilitis at tim t +1? Th Druk Ma has rgaid som cotrol ovr his movmt, ad o logr stays i th sam spot; h oly movs forwards or backwards. Mor formally, lt th Druk Ma s iitial positio b x(0) = 0. Evry scod, h ithr movs forward o pac or backwards o pac, i.., his positio at tim t + 1 will b o of x(t + 1) = x(t) + 1 or x(t + 1) = x(t) 1. W wat to comput th umbr of paths i which th Druk Ma rturs to 0 at tim t ad it is his first rtur, i.., x(t) = 0 ad x(s) 0 for all s whr 0 < s < t. Not, w o logr car about probabilitis. W ar just coutig paths hr. (f) How may paths ca th Druk Ma tak if h rturs to 0 at t = 6 ad it is his first rtur? (g) How may paths ca th Druk Ma tak if h rturs to 0 at t = 7 ad it is his first rtur? (h) How may paths ca th Druk Ma tak if h rturs to 0 at t = 8 ad it is his first rtur? (i) How may paths ca th Druk Ma tak if h rturs to 0 at t = for N ad it is his first rtur? (j) How may paths ca th Druk Ma tak if h rturs to 0 at t = for N ad it is his first rtur? (Hit: rad ad us ay rsult thr if you d.) EECS 70, Fall 2014, Homwork 11 4

5 6. A Idtity o Itgr Partitios Lt b a positiv itgr. A partitio of is a way of writig as a sum of positiv itgrs. Partitios ar cosidrd quivalt udr prmutatio of th summads, so that ordr of th summads dos ot mattr. For xampl, 3 has xactly 3 partitios: 3 = 3 = = W will rprst ach partitio p as a st of pairs (x,r) whr th first lmt x rprsts a summad ad th scod lmt r is th umbr of tims th summad appars, so that w hav = (x,r) p rx for ay partitio p of. W dot by P() th st of partitios of itgr. For xampl: P(3) = {{(3,1)},{(2,1),(1,1)},{(1,3)}} I this problm, w will costruct a combiatorial proof of th followig idtity: For xampl, for = 3, th idtity is sayig: ( ) 1 1!3 1 + p P() (x,r) p ( 1 1! !1 1 1 r!x r = 1 ) ( ) 1 + 3!1 3 = 1 (a) Mak sur th abov idtity works for ay 5. Lt σ b th st of prmutatios ovr {1,2,...,}. Lt f σ. W say that (x 1 x 2... x k ) is a cycl of lgth k of f if ad oly if f (x 1 ) = x 2, f (x 2 ) = x 3,..., f (x k ) = x 1. Not that (x 1 x 2... x k ),(x 2 x 3... x k x 1 ),(x 3 x 4... x 1 x 2 ),... all rprst th sam cycl. A familiar way to rprst a prmutatio f σ is to xplicitly list th mappig (x, f (x)) for all 1 x. A diffrt way to rprst th sam prmutatio is to list all its cycls. Cosidr Tabl 1 for a xampl of this. x f (x) Tabl 1: A prmutatio f σ 7. Th sam prmutatio ca also b rprstd by 1 cycl of lgth 3 ad 2 cycls of lgth 2: (4 6 3), (2 7) ad (1 5). (b) Suppos w ar workig i σ. How may distict cycls of lgth l ca o costruct? (c) Lt (l 1,...,l m ), (x 1,...,x k ), (r 1,...,r k ) b 3 squcs of positiv itgrs such that: i. m i=1 l i =, ii. k j=1 x jr j =, iii. For all j k, thr ar xactly r j distict i m such that l i = x j. How may distict prmutatios i σ ca b rprstd by a st of m cycls of lgth l 1,...,l m? Exprss this umbr oly i trms of ad th rs ad xs. B carful ot to ovr cout prmutatios. EECS 70, Fall 2014, Homwork 11 5

6 (d) You alrady kow that σ =! by a simpl coutig argumt. Now, us th prvious qustio to cout th lmts of σ by usig thir cycl rprstatio i ordr to prov th abov idtity. 7. Fiboacci Fashio You hav accssoris i your wardrob, ad you d lik to pla which os to war ach day for th xt t days. As a Charm School studt, you kow it is t fashioabl to war th sam accssoris multipl days i a row. (Not that th sam gos for clothig itms i gral). Thrfor, you d lik to pla which accssoris to war ach day rprstd by substs S 1,S 2,...,S t, whr S 1 {1,2,...,} ad for 2 i t, S i {1,2,...,} ad S i is disjoit from S i 1. (a) For t 1, prov thr ar F t+2 biary strigs of lgth t with o coscutiv zros (assum th Fiboacci squc starts with F 0 = 0 ad F 1 = 1). (b) Us a combiatorial proof to prov th followig idtity, which, for t 1 ad 0, givs th umbr of ways you ca crat substs of your accssoris for th xt t days such that o accssory is wor two days i a row: 8. Stirlig s Approximatio x 1 0 x 2 0 x t 0 ( )( )( x1 x2 x 1 x 2 x 3 ) ( xt 1 x t ) = F t+2. I this qustio, suppos Z +, w wat to fid approximatios for!. For th parts that ar markd with [VL], plas complt your aswr i th Virtual Lab sklto. You ca also us a oli tool (.g., go to ad typ plot lx ) if you wish to. (a) [VL] Plot th fuctio f (x) = l x. (b) For th followig thr qustios, plas ot that l x is strictly icrasig ad cocav- bcaus, wh x > 0, its first ad scod drivativs ar positiv ad gativ, rspctivly. Cocavity mas that all li sgmts coctig two poits o th fuctio ar blow th fuctio. Suppos Z +, us th plot to xplai why (c) Suppos Z +, us th plot to xplai why l1 + l l l xdx (1) 1 +1 l1 + l l < l xdx (2) 1 (d) Suppos a Z +, us th plot to xplai why ( ) la + l(a + 1) a+1 < l xdx (3) 2 a () Us Equatio ((3)) to prov! ( ). (f) Us Equatio ((4)) to prov! ( ) (Hit: If i this part you fid yourslf wishig you had 1! o th lft-had-sid, try to prov a uppr boud o 1! ad us that to hlp you) (g) Us Equatio ((5)) to prov! ( ), which is a tightr uppr boud. EECS 70, Fall 2014, Homwork 11 6

7 (h) [VL] Th Stirlig s approximatio is usually writt as! 2π ( ) ( or a simplr vrsio! ). 2π( ) Plot th fuctio f () =!. What do you obsrv? (i) Suppos m = k, us m, ad apply th simplr vrsio of th Stirlig s approximatio to rwrit ( k). (j) [VL] Now, suppos m 1 = k 1 = 0.25, m 2 = k 2 = 0.5, ad m 3 = k 3 = 0.75, plot l( ( ) ( k 1 ), l( ) k 2 ), ad l( ( ) k 3 ) as fuctios of o a plot with liar-scald axs. What do you obsrv? 9. Writ your ow problm Writ your ow problm rlatd to this wk s matrial ad solv it. You may still work i groups to braistorm problms, but ach studt should submit a uiqu problm. What is th problm? How to formulat it? How to solv it? What is th solutio? EECS 70, Fall 2014, Homwork 11 7