random walk

[MattcAnderson]

Hi again MersenneForum,

The problem seems to have not been stated clearly. Suppose a starting point at a point on a Euclidean plane. Label this point the origin. Then make two axis. Label them North and East.

The creature will 'go' North whenever it flips heads on a fair coin. It will go East whenever it flips tails on the same coin. There are an infinite number of 'go' moves. The next distance is always half the last distance.

The question is - where will the creature probably end up?

Hint - It must be in the North East quadrant.

Now I believe I have a more complete solution. The line N = 2 - E, where N is for North and E is for East, contains all possible end points. Specifically, 0 <= E <= 2 are the possible locations of convergence. Since it is a fair coin, most positions along this line, within the limits, are equally likely.
So it will be a normal distribution on the line.

Many thanks to the contributions so far. The next question is how can one describe mathematically, the possible end locations for our creature with a random walk? It can only land on rational points in the plane.

Do you like my animated gif?

Regards,
Matt

[Uncwilly]

Originally Posted by MattcAnderson

The next question is how can one describe mathematically, the possible end locations for our creature with a random walk? It can only land on rational points in the plane.

I think you expressed that in your spoiler.

[LaurV]

Originally Posted by MattcAnderson

It can only land on rational points in the plane.

Are you sure? Can you prove that? (this is a tricky question... one could easily prove the contrary)
(and if so, are all rational points on the line touched? or some of them can not be touched?)
(or are any real, non-rational points there? which one? how many)

Just for fun, here is some additional play with it, it shows few random paths, with the end point marked red, and the line where all the paths ends. The traveled distance is always \(2\), and the displacement is anything between \(\sqrt 2\) and \(2\). For safety, there is no macro, so one has to hold F9 down for random animation.

drunkman.zip