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An interesting random walk problem

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  • vkg378
    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
    Message 1 of 3 , Apr 1, 2003
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      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.
    • jason1990
      ... below ... 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
      Message 2 of 3 , Apr 3, 2003
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        --- In probability@yahoogroups.com, "vkg378" <vkg378@y...> wrote:
        > 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.

        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.
      • jason1990
        Sorry, I swapped the roles of k and N. ... comes ... ends
        Message 3 of 3 , Apr 3, 2003
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          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
          > 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.
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