## One dimensional random walk

Posted by peeterjoot on February 3, 2013

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## Question: One dimensional random walk

Random walk in 1D by unit steps. With the probability to go right of and a probability to go left of what are the first two moments of the final position of the particle?

## Answer

This was a problem from the first midterm. I ran out of time and didn’t take the answer as far as I figured I should have. Here’s a more casual bash at the problem.

First we need the probabilities.

**One step: **

Our distance (from the origin) can only be .

**Two steps: **

We now have three possibilities

**Three steps: **

We now have three possibilities

**General case**

The pattern is pretty clear, but we need a mapping from the binomial index to the the final distance. With an index , and a guess

where

So

and

Our probabilities are therefore

**First moment**

For the expectations let’s work with instead of , so that the expectation is

This gives us

**Second moment**

So the second moment is

From this we see that the variance is just this second term

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