riddles on a line [#2]

A second Riddle(r), with a puzzle related with the integer set Ð={,12,3,…,N}, in that it summarises as

Given a random walk on Ð, starting at the middle N/2, with both end states being absorbing states, and a uniform random move left or right of the current value to the (integer) middle of the corresponding (left or right) integer interval, what is the average time to one absorbing state as a function of N?

Once the Markov transition matrix M associated with this random walk is defined, the probability of reaching an absorbing state in t steps can be derived from the successive powers of M by looking at the difference between the probabilities to be (already) absorbed at both t-1 and t steps. From which the average can be derived.

avexit <- function(N=100){
 #transition matrix M for the walk
 #1 and N+2 are trapping states
 tranz=matrix(0,N+2,N+2)
 tranz[1,1]=tranz[N+2,N+2]=1
 for (i in 2:(N+1))
  tranz[i,i+max(trunc((N+1-i)/2),1)]=tranz[i,i-max(trunc((i-2)/2),1)]=1/2
 #probabilities of absorption
 prowin=proloz=as.vector(0)
 init=rep(0,N+2)
 init[trunc((N+1)/2)]=1 #first position
 curt=init
 while(1-prowin[length(prowin)]-proloz[length(prowin)]>1e-10){
  curt=curt%*%tranz
  prowin=c(prowin,curt[1])
  proloz=c(proloz,curt[N+2])}
 #probability of new arrival in trapping state
 probz=diff(prowin+proloz)
 return(sum((2:length(proloz))*probz))}

leading to an almost linear connection between N and expected trapping time.

One Response to “riddles on a line [#2]”

  1. […] leave a comment for the author, please follow the link and comment on their blog: R – Xi'an's Og. R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data […]

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.