From the above tables we observe that as \(\Delta t\) becomes smaller, the variance of the sample of the final position becomes closer to the variance predicted by the model which is \(2Dt\).
The mean remains the same which is \(\beta t\).
We observe that if the total walk time is large (experiment #4) , then more steps are needed to bring \(\Delta t\) to be small enough so that the variance becomes close to \(2Dt\).\(.\) This answers the second question we are set to solve in this project which is Does the variance of the above distribution converges, as \(\Delta t\rightarrow 0\) and \(\Delta x\rightarrow 0\) under the above mentioned condition of keeping \(\frac {\left ( \Delta x\right ) ^{2}}{\Delta t},\) to the analytical variance of \(2Dt\) and the theoretical mean of \(\beta t\)?
Now to answer the first question of convergence of the histogram of the final position to the normal.
Looking at the quantile plots we observe that as more steps are used (hence smaller \(\Delta t\) and smaller \(\Delta x\)) then the quantile-quantile plot was tilting closer and closer to the straight line at \(45^{0}\) which would be the case when we plot the quantile of 2 data sets coming from the same distribution. This concludes that the final distribution of the random walk position converges to normal distribution with the above parameters.
The following diagram below shows a run where on the left side there is a plot showing the quantile plot when the number of steps is small. The plot on the right side shows the quantile plot at the end of the run when \(n\) was large. We see that the quantile plot line is now almost exactly over the \(45^{0}\) line, confirming that the data is coming from normal distribution.
Therefore, we have answered the 2 questions this simulation was designed to answer.