- Hi,

I have an interesting problem on the random walk. If anybody takes

time to give me some hints or may be a solution I would be grateful.

Consider a one dimensional random walk with unit steps. The

pobability of forward and backward steps is 0.5 each. It is given

that random walk starts at 0 and ends at a given integer k > 0 at

time N. What is the probabilty that the random walk never goes below

zero at anytime from 0 to N. - --- In probability@yahoogroups.com, "vkg378" <vkg378@y...> wrote:
> Hi,

below

> I have an interesting problem on the random walk. If anybody takes

> time to give me some hints or may be a solution I would be grateful.

>

> Consider a one dimensional random walk with unit steps. The

> pobability of forward and backward steps is 0.5 each. It is given

> that random walk starts at 0 and ends at a given integer k > 0 at

> time N. What is the probabilty that the random walk never goes

> zero at anytime from 0 to N.

Try the reflection principle. Suppose we have a path that starts at

0, makes an initial step in the positive direction, eventually comes

back to 0 at a time T < k, then goes to N at time k. We can "flip"

the path between times 0 and T to get a path that goes from 0 to N

with an initial step in the negative direction and first hits 0 at

time T. But *any* path that starts in the negative direction and ends

at N must hit 0 at some point.

I'll leave it for you to work it out from here. - Sorry, I swapped the roles of k and N.

--- In probability@yahoogroups.com, "jason1990" <jason1990@y...>

wrote:> Try the reflection principle. Suppose we have a path that starts at

comes

> 0, makes an initial step in the positive direction, eventually

> back to 0 at a time T < k, then goes to N at time k. We can "flip"

ends

> the path between times 0 and T to get a path that goes from 0 to N

> with an initial step in the negative direction and first hits 0 at

> time T. But *any* path that starts in the negative direction and

> at N must hit 0 at some point.

>

> I'll leave it for you to work it out from here.