The Random Walk For Dummies

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1 The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli process ad the Catala umbers i greater depth Fially we determie the probability that, if a druk is foud agai at the bar, the this is his first retur visit Itroductio The radom walk has bee a topic of iterest i may disciplies, but it has bee of particular iterest i probability theory Ideed, every studet of probability theory has heard of the radom walk, especially i the form of a druk leavig a bar ad waderig aimlessly up ad dow the street Amog other issues, the followig four have bee ivestigated i oe, two, ad three dimesios: The expected positio x of the druk after steps The maximum positio x that the druk has reached after steps The expected time of the druk s last visit to The probability that the druk has t stumbled upo his ow path after steps These issues are discussed, for example, i Rota s book o probability [3] The oe-dimesioal discrete case is most widely kow because it is simple, yet illustrates may iterestig ad importat features I this paper, we treat this case of the first issue above Thus we aswer a basic questio: What is the probability that the druk is at a certai distace x, from the bar, after steps? I Sectio, we review five basic topics: the sample space, the biomial coefficiet, radom variables, the Beroulli process, ad Catala umbers I Sectio 3, we fid the probability distributio for the positio of the druk after steps I Sectio, we proceed to calculate the probability that the druk arrives at the bar for the first time after steps Fially, i Sectio 5, we determie the coditioal probability that the druk s first retur occurs o the th step give that he is ideed at the bar the; the formula is surprisigly simple The Basics I this sectio, we review five cocepts of probability theory, which we will use to study the radom walk of the druk First, a sample space Ω is defied to be the set of all possible outcomes of a experimet Cosider the coi toss as a example A sigle toss has two possible outcomes: heads H, or tails T ; thus Ω = {H, T } If we toss the coi twice, the there are differet possibilities; ow Ω = {T T, T H, HT, HH} Secod, the biomial coefficiet ( k) is defied to be the umber of k-combiatios of a -elemet set I other words, it is the umber of differet ways to pick k elemets out of The biomial coefficiet is give by the formula, ( )! = k ( k)! k! 3

2 MIT Udergraduate Joural of Mathematics This formula is proved i [, p 6] Third, a radom variable is defied to be a fuctio X that assigs to each elemet c, i the sample space of a experimet, oe ad oly oe real umber X(c) The sample space Ω of X is the set of real umbers x such that x = X(c) for some c i the sample space of the experimet This defiitio is foud i [, p ] Fourth, a Beroulli process is defied to be a sequece of radom variables X, X, X 3, Each X i records the outcome of a experimet modeled by the toss of a coi Let p equal the probability of gettig a head, ad q the probability of gettig a tail Sice these are the oly possible outcomes, p + q = Let X i be if the th trial yields a head, ad be if a tail Thus, the sample space Ω of a Beroulli process is the set of all possible sequeces of ad Fifth ad fially, the th Catala umber c couts certai arragemets of paretheses A well-formed arragemet is a list of paretheses where each ope parethesis ca be paired with a correspodig closed parethesis to its right The th Catala umber is the umber of such well-formed arragemets For example, whe = 3, the possible well-formed arragemets are ()()(), ()(()), (())(), (()()), ((())) Accordig to [, p 53], the th Catala umber c is give by the formula, c = ( ) + Thus, for example, the first six Catala umbers are,,, 5,, ad 3 How It All Ties I Suppose a druk leaves a bar ad walks aimlessly up ad dow the street, totally disorieted We model the street as a lie with the bar at the origi, ad assume that the druk takes uit steps, so we may record his positio with a iteger Thus, for example, if he takes 5 steps to the left, he will be at positio 5 We ow calculate the probability that the druk is at positio x after steps Assume that the druk s walk ca be modeled by a Beroulli process Say that the ith step is represeted by the radom variable X i A value of idicates a step to the left; a value of, a step to the right, so that G = X + X + + X gives the positio of the druk after steps We wat to kow the distributio of the radom variable G Deote the value of G by x Deote the umber of steps take to the right by r, the umber to the left by l The It follows that x = r l ad = r + l r = (x + ) ad l = ( x) Now, there are ( l) ways that l give steps ca occur amog total steps This is also the umber of ways of arrivig at the poit x, ad each way has probability p r q l Note that ad x must have the same parity because x = l We ca therefore coclude that the probability distributio at poit x is give by the formula,

3 The Radom Walk For Dummies 5 { ( ) P (G = x) = l p r q l, if x = mod ;, otherwise Cosider for example the case p = q = ; this is the case of the symmetric radom walk The probability P (G = x) of beig at positio x after steps is give i Table 3- This table is simply a pascal triagle iterspersed with s Table 3- The Symmetric Radom Walk \ x The probability distributio i this particular case is give by ( )( ) ( )/ P (G = x) = = l l Sice every path is equally likely i the symmetric radom walk, the probability ca be iterpreted purely combiatorially The probability is the umber of differet ways of arrivig at x divided by the size of the sample space The sample space is the set of all the possible paths of legth Sice the druk has two choices at each poit, ad he takes a total of steps, the total umber of possibilities is Takig a Step Further From ow o, we assume that the radom walk is symmetric; that is, p = What is the probability that the druk s first retur is at the th step? Here is where Catala umbers eter Observe that there is a oe-to-oe correspodece betwee paths ad arragemets of paretheses First, let a ope parethesis represet a step to the left, ad a closed parethesis a step to the right For ow, we cosider oly the case where the druk s first step is towards the left, sice the case to the right is clearly symmetric to it From Sectio, recall the defiitio of the Catala umber c : it is the umber of well-formed arragemets of paretheses Note that, for well-formed arragemets, the umber of ope paretheses is always at least that of closed paretheses, regardless of what first k parethesis we pick Because of the correspodece betwee paths ad arragemets, the Catala umbers cout the umber of paths that the druk ca take, which start ad ed at the bar, without ever crossig to the right side of the bar We ow adapt our correspodece to the problem of the druk s first retur The druk takes his first step to the left O the ext steps, we isist that the druk s path correspods to a well-formed arragemet so that at step he is at positio

4 6 MIT Udergraduate Joural of Mathematics without ever crossig to the right of The the fial step brigs the druk back ito the bar This couts the total umber of differet paths that the druk ca take give the coditio that the first retur be at the th step Summarizig, we have the followig sequece of steps: Step The druk takes oe step to the left The ext - steps The druk follows a path, which correspods to a well-formed arragemet, that restricts the druk to be either at poit or to the left of it, ad that forces him to be at after the ( )st step Step The druk s first retur visit occurs whe he takes oe step to the right ad ito the bar Hece, the umber of paths the druk ca take is simply the Catala umber c, sice the well-ordered restrictio applies oly to the middle steps We kow that the th Catala umber is equal to + ( ) So the ( )st Catala umber is simply ( ) Allowig the druk to take his first step to the right will double the total umber of paths that the druk ca take So the total is ( ) To compute the probability of first retur, we eed oly divide this quatity by the size of the total sample space The latter was see at the ed of Sectio 3 to be sice there are steps Therefore, the probability that the druk reaches the bar for the first time after steps is ( ) / So, for example, cosider the probability that the first retur is at Step = 6; this probablity is give by ( )/ 6 = Coditioal Probability ad the First Retur We ow cosider a variat of the problem studied i the previous sectio: Give that the druk is at the bar at Step, what is the probability that this is his first retur visit? This problem is like the previous oe The oly differece is that the sample space has bee reduced Istead of cosiderig all of the possible paths, we ow cosider the total umber of paths give that, i the ed, the druk will be at the bar, which he may or may ot have passed by earlier

5 The Radom Walk For Dummies 7 To fid the probability that the druk will be at the bar after steps, we use the coclusios from Sectio 3 to obtai ( ) P (G = ) = p () q () Sice p = q =, this expressio becomes ( ) / The umerator i this expressio represets the umber of paths the druk ca take provided he is at the bar at the th step This is sample space we eed for the coditioal probability From Sectio, recall that, if the druk s first retur visit is o the th step, the the umber of paths that the druk ca take is ( ) O the other had, i Sectio 3, we foud the umber of paths that the druk ca take that put him back at the bar o Step ; this umber is simply ( ) Fially we divide these two umbers, gettig ( ) / ( ) = / ( )! ()! ( )!( )!!!, which simplifies to / ( ) = ( ) = The fial simplicity is amazig! The formula is just ad is uexpectedly simple! #(steps), Refereces [] Brualdi, R, Itroductory Combiatorics, Pretice-Hall, Third Editio, 999 [] Hogg, Robert V, Itroductio to Mathematical Statistics, Fifth Editio, 995 [3] Rota, G C, Probability Theory, Prelimiary Editio, 99

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