Question

In: Statistics and Probability

Let b > 0 be an integer. Find the probability that a symmetric simple random walk...

Let b > 0 be an integer. Find the probability that a symmetric simple random walk started from 0 visits b the first time in the nth step.

Hint: Draw a picture, and try to describe the requirements that the path consisting the first n−1 steps should satisfy. The Reflection principle (or a related result) should be helpful after that.

Expert Solution

A random walk is symmetric if the probability of going to each of the neighbors is the same.

Since the probability has been asked for visiting b for the first time, hence we would first calculate the probability of visiting b-1 in n-1 steps and then multiply it with 1/2 to get the required probability.

So let us first calculate the probability of visiting b-1 in n-1 steps

Now, we will visit b-1 if the difference between the number of forward and backward steps is b-1.

Hence, we can write

Hence, we get

and

Now, the total number of ways in which we can have and steps in a total of steps

In order to calculate the probability, we need to know the sample space

So, the total number of ways possible in steps

Hence, the probability of visiting in steps ,denoted by

Now, the probability of visiting b in n steps

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n forward = (n +b-2)/2

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n forward

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orchestra answered 1 year ago

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Question

Let b > 0 be an integer. Find the probability that a symmetric simple random walk...

Solutions

Expert Solution

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