Grimmett and Stirzaker Exercise 3.11.23

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Consider a symmetric random walk with:



Absorbing barrier at N

Reflecting barrier at 0, (so that , when the particle is at zero, it moves 1 at the next step)

$alpha_k(j)$ probability that the particle, having started at k, visits 0 exactly j times before being absorbed at N. If k=0 then the starting point counts as one visit.



It has been shown previously that the probability that the particle hits 0 before N is $(1-frackN)$. Therefore



$P(M geq r|S_0=k)=(1-frackN)(1-frac1N)^r-1$



This much is straightforward.



Then:
beginalign
P(M = j|S_0=k)&=P(M geq j|S_0=k)-P(M geq j+1|S_0=0) \
&=(1-frackN)(1-frac1N)^j-1 - (1-frac1N)^j+1\
&=(1-frackN)(1-frac1N)^j-1frac1N
endalign



Is the second line above correct?



If so how so they go from the second line to the third line?




probability




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    up vote
    1
    down vote

    favorite












    Consider a symmetric random walk with:



    Absorbing barrier at N

    Reflecting barrier at 0, (so that , when the particle is at zero, it moves 1 at the next step)

    $alpha_k(j)$ probability that the particle, having started at k, visits 0 exactly j times before being absorbed at N. If k=0 then the starting point counts as one visit.



    It has been shown previously that the probability that the particle hits 0 before N is $(1-frackN)$. Therefore



    $P(M geq r|S_0=k)=(1-frackN)(1-frac1N)^r-1$



    This much is straightforward.



    Then:
    beginalign
    P(M = j|S_0=k)&=P(M geq j|S_0=k)-P(M geq j+1|S_0=0) \
    &=(1-frackN)(1-frac1N)^j-1 - (1-frac1N)^j+1\
    &=(1-frackN)(1-frac1N)^j-1frac1N
    endalign



    Is the second line above correct?



    If so how so they go from the second line to the third line?




    probability




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Consider a symmetric random walk with:



      Absorbing barrier at N

      Reflecting barrier at 0, (so that , when the particle is at zero, it moves 1 at the next step)

      $alpha_k(j)$ probability that the particle, having started at k, visits 0 exactly j times before being absorbed at N. If k=0 then the starting point counts as one visit.



      It has been shown previously that the probability that the particle hits 0 before N is $(1-frackN)$. Therefore



      $P(M geq r|S_0=k)=(1-frackN)(1-frac1N)^r-1$



      This much is straightforward.



      Then:
      beginalign
      P(M = j|S_0=k)&=P(M geq j|S_0=k)-P(M geq j+1|S_0=0) \
      &=(1-frackN)(1-frac1N)^j-1 - (1-frac1N)^j+1\
      &=(1-frackN)(1-frac1N)^j-1frac1N
      endalign



      Is the second line above correct?



      If so how so they go from the second line to the third line?




      probability




      share|cite|improve this question











      Consider a symmetric random walk with:



      Absorbing barrier at N

      Reflecting barrier at 0, (so that , when the particle is at zero, it moves 1 at the next step)

      $alpha_k(j)$ probability that the particle, having started at k, visits 0 exactly j times before being absorbed at N. If k=0 then the starting point counts as one visit.



      It has been shown previously that the probability that the particle hits 0 before N is $(1-frackN)$. Therefore



      $P(M geq r|S_0=k)=(1-frackN)(1-frac1N)^r-1$



      This much is straightforward.



      Then:
      beginalign
      P(M = j|S_0=k)&=P(M geq j|S_0=k)-P(M geq j+1|S_0=0) \
      &=(1-frackN)(1-frac1N)^j-1 - (1-frac1N)^j+1\
      &=(1-frackN)(1-frac1N)^j-1frac1N
      endalign



      Is the second line above correct?



      If so how so they go from the second line to the third line?





      probability





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 18 at 14:28









      Bazman

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