Erich Prisner

## Random Walks on a path of length 4

Try the random walk on a path of length 4.

Here the probability distributions:

Let a(n), b(n), c(n), and d(n) denote the probabilities that the the fields
are visited in the nth step. If the right field is absorbing, the recurrence
formulas are

a(n)=b(n-1)/2 |
b(n)=an+c(n-1)/2 |

c(n)=b(n)/2 |
d(n)=c(n)/2 |

with initial conditions a(0)=1, b(0)=0, c(0)=0, and d(0)=0.

When solving this system, we get.
The characteristic equation for the resulting recurrence relation, -4x^{2}+3=0
has two zeros, ,and ,
and when solving the system of linear equations we obtain, we get

d(n)=
.

Obviously for even n we have d(n)=0, and for odd n we get d(n)=.

For the expected value for the transition time from the leftmost to the rightmost
cell, (using a formula for a series) we get

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Erich Prisner 2004