A number of experiments were performed for different input parameters. The table below lists the variance of the distribution of the final position as the number of steps is increased. The run parameters are also shown
starting step number\(=2\), \(\beta =2,t=2,D=3,final\ p=0.557,final\ q=0.443\)
sample size \(5000,\) number of bins \(40,\) seed\(=123456\)
| \(n\) (number of steps) | Variance | True variance (2Dt) | \(\Delta t\) |
| \(2\) | \(3.92\) | \(12\) | \(1\) |
| \(7\) | \(9.73\) | \(12\) | \(0.2857\) |
| \(12\) | \(10.43\) | \(12\) | \(0.1667\) |
| \(17\) | \(10.9\) | \(12\) | \(0.1176\) |
| \(22\) | \(11.37\) | \(12\) | \(0.0909\) |
| \(27\) | \(11.19\) | \(12\) | \(0.0741\) |
| \(32\) | \(12.02\) | \(12\) | \(0.0625\) |
| \(\cdots \) | \(\cdots \) | \(\cdots \) | \(\cdots \) |
| \(67\) | \(12.05\) | \(12\) | \(0.0299\) |
| \(72\) | \(11.89\) | \(12\) | \(0.0278\) |
| \(77\) | \(12.16\) | \(12\) | \(0.0260\) |
| \(82\) | \(11.99\) | \(12\) | \(0.0244\) |
| \(87\) | \(11.78\) | \(12\) | \(0.0230\) |
| \(92\) | \(12.03\) | \(12\) | \(0.0217\) |
| \(97\) | \(11.88\) | \(12\) | \(0.0206\) |
| \(102\) | \(11.47\) | \(12\) | \(0.0196\) |
Since the parameters \(t,D,\beta \), then running for \(n=50\) will produce the same numerical values already contained in the first experiment when looking at the table above up to \(n=50\,\) (the program starts by seeding the random number generator, so nothing will change here and we will just produce a subset of the result already produced in first experiment). So I will just show the final plot, showing the convergence of the histogram and the quantile-quantile plot
Again, as described at the start of experiment 2 above, this is a subset of the first experiment. We will show the final plot only to show how close to the standard normal the final position histogram is.
The following 2 experiments are not required to do, but they are extra experiments I already done and included here.
starting step number\(=400\), \(\beta =5,t=100,D=3,final\ p=0.623,final\ q=.377\)
sample size \(5000,\) number of bins \(60,\) seed\(=123456\)
final \(\Delta x=0.2945\) final \(\Delta t=0.0145\)
| Experiment number | \(n\) (number of steps) | Variance | True variance (2Dt) | \(\Delta t\) |
| \(1\) | \(400\) | \(1.89\) | \(600\) | \(0.2392\) |
| \(2\) | \(900\) | \(340\) | \(600\) | \(0.1089\) |
| \(3\) | \(1400\) | \(420\) | \(600\) | \(0.0705\) |
| \(4\) | \(1900\) | \(464\) | \(600\) | \(0.0521\) |
| \(5\) | \(2400\) | \(504\) | \(600\) | \(0.0414\) |
| \(6\) | \(2900\) | \(514\) | \(600\) | \(0.0343\) |
| \(7\) | \(3400\) | \(525\) | \(600\) | \(0.0293\) |
| \(8\) | \(3900\) | \(546\) | \(600\) | \(0.0255\) |
| \(9\) | \(4400\) | \(536\) | \(600\) | \(0.0226\) |
| \(10\) | \(4900\) | \(533\) | \(600\) | \(0.0203\) |
| \(11\) | \(5400\) | \(552\) | \(600\) | \(0.0185\) |
| \(12\) | \(5900\) | \(558\) | \(600\) | \(0.0169\) |
| \(13\) | \(6400\) | \(567\) | \(600\) | \(0.0156\) |
| \(14\) | \(6900\) | \(583\) | \(600\) | \(0.0145\) |
starting step number\(=5,\beta =5,t=1,D=3,final\ p=0.579,final\ q=0.421\)
sample size \(5000,\) number of bins \(50,\) seed\(=123456\)
final \(\Delta x=0.1907\), final \(\Delta t=0.0061\)
| Experiment number | \(n\) (number of steps) | Variance | True variance (\(2Dt\)) | \(\Delta t\) |
| \(1\) | \(5\) | \(1.019\) | \(6\) | \(0.2\) |
| \(2\) | \(10\) | \(3.4\) | \(6\) | \(0.1\) |
| \(3\) | \(15\) | \(4.09\) | \(6\) | \(0.0667\) |
| \(4\) | \(20\) | \(4.74\) | \(6\) | \(0.05\) |
| \(5\) | \(25\) | \(5\) | \(6\) | \(0.4\) |
| \(6\) | \(30\) | \(5.18\) | \(6\) | \(0.0333\) |
| \(7\) | \(35\) | \(5.43\) | \(6\) | \(0.0286\) |
| \(8\) | \(40\) | \(5.466\) | \(6\) | \(0.0250\) |
| \(9\) | \(45\) | \(5.3\) | \(6\) | \(0.0222\) |
| \(10\) | \(50\) | \(5.66\) | \(6\) | \(0.02\) |
| \(11\) | \(55\) | \(5.4\) | \(6\) | \(0.0182\) |
| \(12\) | \(60\) | \(5.85\) | \(6\) | \(0.0167\) |
| \(\cdots \) | \(\cdots \) | \(\cdots \) | \(\cdots \) | \(\cdots \) |
| \(31\) | \(150\) | \(5.78\) | \(6\) | \(0.0065\) |
| \(32\) | \(155\) | \(5.909\) | \(6\) | \(0.0063\) |
| \(33\) | \(160\) | \(5.75\) | \(6\) | \(0.0061\) |