Consider a particle that moves along the set of integers in the following manner If it...

 

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Consider a particle that moves along the set of integers in the following manner If it is presently at i then it next moves to i + 1 with probability p and to i - 1 with probability 1 - p Starting at 0, let a denote the probability that it ever reaches 1. (a) Argue that a = p + (1 - p)a'. (b) Show that if p> 1/2 if p< 1/2 1 a p/(1-p) (c) Find the probability that the particle ever reaches n, n > 0 (d) Suppose that p < 1/2 and also that the particle eventually reaches n, n > 0 lf the particle is presently at i, i < n, and n has not yet been reached, show that the particle will next move toi +1 with probability 1 - p and to i - 1 with probability p That is, show that P{next at i + 1 at i and will reach n} = 1 - p %3D (Note that the roles of p and 1 given that n is eventually reached) p are interchanged when it is Consider a particle that moves along the set of integers in the following manner If it is presently at i then it next moves to i + 1 with probability p and to i - 1 with probability 1 - p Starting at 0, let a denote the probability that it ever reaches 1. (a) Argue that a = p + (1 - p)a'. (b) Show that if p> 1/2 if p< 1/2 1 a p/(1-p) (c) Find the probability that the particle ever reaches n, n > 0 (d) Suppose that p < 1/2 and also that the particle eventually reaches n, n > 0 lf the particle is presently at i, i < n, and n has not yet been reached, show that the particle will next move toi +1 with probability 1 - p and to i - 1 with probability p That is, show that P{next at i + 1 at i and will reach n} = 1 - p %3D (Note that the roles of p and 1 given that n is eventually reached) p are interchanged when it is

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It is given that particle moves along the set of integers If it is present at i then it next moves t... View the full answer