Random walks

Number of random walks: 1 2 3 4 5 6 7 8 9 10 15 20 30 50 100 200

Fixed values in this simulation are:

• stepsize [math]\displaystyle{ {}=5 }[/math] and
• Number of steps per walk [math]\displaystyle{ {}= 100 }[/math].

Therefore, the expected squared distance from the starting point will be equal to

• [math]\displaystyle{ 100\cdot 5^2=2500 }[/math].

Average square of the distance between starting point and endpoint of the walks:

Source code

<jsxgraph width="600" height="600">
var brd = JXG.JSXGraph.initBoard('jxgbox', {originX: 300, originY: 300, unitX: 3, unitY: 3});
var t = brd.createElement('turtle');

function run() {
  var i,j,dist,sumdist=0.0;
  var stepSize = 5; 
  brd.suspendUpdate();
  var nr = document.getElementById('number').value*1;
  for (i=0;i<nr;i++) {
     t.setPenColor(JXG.hsv2rgb(Math.round(Math.random()*255),Math.random(),Math.random()));
     for (j=0;j<100;j++) {
        var a = Math.floor(360*Math.random()); 
        t.right(a); 
        t.forward(stepSize);
     }
     dist = t.pos[0]*t.pos[0]+t.pos[1]*t.pos[1];
     sumdist += dist;
     t.home();
  }
  document.getElementById('output').value = (sumdist/nr).toFixed(3);
  brd.unsuspendUpdate();
}
function clearturtle() {
  t.cs();
}
</jsxgraph>

External links

• http://en.wikipedia.org/wiki/Random_walk
• http://mathworld.wolfram.com/RandomWalk.html
• http://www.bookrags.com/research/random-walks-and-brownian-motion-wop/