You are Here: Home >< Maths

# 2d random walk watch

1. Hi,
I'm stuck with this question about random walks on Z^2

Let S_n = (X_n, Y_n) be a simple symmetric random walk in Z^2, starting from (0, 0), and set T = inf {n >= 0: max{|X_n|, |Y_n|} = 2}.Calculate E(T) (Expectation of T).

Michelle (Mic)
2. anyone?
3. There aren't actually many possibilities: there are only 4 states you need to consider

A: particle is at the origin
B: particle is distance 1 from the origin
C: particle is distance sqrt(2) from the origin.
D: particle is >=2 from the origin (so max|X_n|,|Y_n| = 2).

Write T_B for inf {n >= 0: max{|X_n|, |Y_n|} = 2} when you start from state B (and T_C, T_D similarly. Clearly T_D = 0).

If you think about what happens if you take one "step" from state A, it's obvious that T = T_B + 1.
Now think about what happens if you take one "step" from state B. What's the chance you end up in state A? State C? State D? So we can write a linear equation relating T, T_B, T_C.
Do the same for state C.

So, 3 equations, 3 unknowns. Solve for T, T_B, T_C.
4. ok, I understand this, but where does this lead to? Having known these, how would i compute E(T)?

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: February 6, 2010
The home of Results and Clearing

### 3,509

people online now

### 1,567,000

students helped last year
Today on TSR

### IT'S TODAY!

A-level results chat here

### University open days

1. Bournemouth University
Fri, 17 Aug '18
2. University of Bolton
Fri, 17 Aug '18
3. Bishop Grosseteste University
Fri, 17 Aug '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams