The Very Basics of the Random Walk

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It’s cited and used in a great many disciplines, however you’ll find it most often used in the study of probability.  It’s often one of the first lessons a probability student will hear, the story of a drunk leaving a bar and attempting to head off home.

There are lots of areas to consider when we are investigating the drunk’s walk.

  • What position do we estimate the drunk will be after n steps?
  • Expected time of the drunks last visit to 0
  • the maximum position of x that the drunk may reach after n steps
  • the probability that the drunk hasn’t backtracked after n steps

The example isn’t a difficult one, yet it illustrates many important questions about probability.  The main question is how far did the drunk get from the bar or to phrase it more specifically – what is the probability that the drunk is at a specific point x after a certain number of drunken steps n.

There are several interesting mathematical concepts to cover in this question.  We have to look at the sample space, basic probability, the random variables involved and eveb the Bernouilli principle.  We can work out a full probability distribution for the final position of the drunk, or at least after he has taken n number of steps!

So how do we even start to calculate this.  Well first we have to make a few assumptions, the drunk has to take unit steps for example mainly so we can estimate his steps with an integer.  We also have to presume that the position can be recorded from a straight line from the bar to the end position.  This means for example if he takes 4 steps to the left then we can state that the position is -4, then steps to the right would be +1 so 3 steps to the right would addd +3 to the location.

So we can then model on the basis of the above equation which will give the position of the drunk after taking n steps.  the random variable is of course Gn and to work out the possibilities you need to work out the distribution of that variable.

You can of course take this a lot further, and there’s lots of great websites that will explain this concept in more detail.  try the BBC website whch has some great tutorials and radio programmes covering similar concepts – posted here on how to access.  There’s also some papers and documents available on the websites of big universities.  I found some interesting ones on a few Polish University sites but last time I checked you needed a Polish IP address to access them for some unknown reason, there’s a link here.