Random Walk in 1D: Finding Average Steps to the Right After N Steps

[ak416]

Random walk in one dimension. A person (say, in an unstable state of mind/body) is moving in
one dimension, with coordinate x, starting at x = 0. Assume: i.) that s/he moves in steps of length
l, ii.) that the probability that s/he takes a step to the left is p, while the probability of taking a
step to the right is q = 1 − p and iii.) that all the steps are independent (i.e. the probability of
taking the n + 1-th step left or right is independent on what the previous n steps were).

One of the questions ask: Find the average number of steps to the right, <nR>, taken after N steps.

This is what I got:
<nR> = sum i=0toN i*(N choose i) * p^(i) * (1-p)^(N-i)

A played around with it but i can't seem to get it into a nicer form. Other questions then ask for the variance and to compare it to the mean, so I am sure i have to somehow eliminate the summation sign. There's a hint saying to use the fact that p d/dp(p^n) = np^n. Is there a way to simplify this?

 

[0rthodontist]

Well, the way you set it up (if you're sure that's what the question is asking) that is just the mean of a binomial distribution with parameters N and p.

 

[ak416]

ya i know, i used the binomial distribution formula to come up with that formula. I think that's the best i can do. I have a better question. What is the limit as n approaches infinity? Because one of the other questions asks what the large N limit is for the ratio Var(nR) / <nR> . I managed to derive the formula for Var(nR) to be p*d/dp(<nR>) - <nR>^2 , so the limit I am looking for is (p*d/dp(<nR>))<nR> - <nR> , so as a start I would like to find the limit for large N of this <nR>. Anyone?

 

[ak416]

my guess is p * N but how do i derive that mathematically?

 

[ak416]

actually i think i messed up on the d/dp part, forgot to take into account that 1-p depends on p! If anyone can help me figure out a nice formula for the variance that would be great.

 

[0rthodontist]

Well, it is p * N and the variance is Np(1 - p). John E. Freund's Mathematical Statistics gives proofs, but your book also probably does that. Never mind; I withdraw from this.

 

[ak416]

Ya thanks, i was looking through my stats notes and saw the derivations. I am going to try finishing up the other problems now. Assignment is due in the morning!