We resume our discussion on Monte Carlo Simulation of Synthetic Data Sets shortly. This is a powerful technique. It assumes that if the fitted parameters a0 is a reasonable estimate of the true parameters by minimizing the chi-square then the distribution of difference in subsequent parameters from a0 should be similar to that of the corresponding calculation with true parameters
The assumed a0 initial parameter set together with noise can be used to generate new data sets at the same values of the independent  axis coordinate as the actual data set has with that axis. The synthetic data aligns with the actual data at these points of reference on the line. Now the interesting part is that the new dataset has the same relationship to a0 as the actual data has with the true parameter set. 
We can generate many such data sets and they are called synthetic data sets. For each such dataset, we can fit a model and obtain the corresponding parameter set. This yields one data point for the difference between the parameter set  corresponding to the synthetic data from the initial parameter set. The simulation can be run thousands of times to generate sufficient data points. This leads to a probability distribution which is M-dimensional.  We can even calculate the standard deviation of the original parameter set after fit using the cumulative data point differences.
#codingexercise 
Given two natural number n and m, find the number of ways in which the numbers that are greater than or equal to m can be added to get the sum n.
for example, n = 3 and m = 1
We we have three ways : 1 + 1 + 1, 1 + 2, 3
int GetCountWays(int n, int m)
{
    int [,] dp = new int[n+2, n+2];
    dp [0, n+1] = 1;

    for (int k = n; k > m; k--) {
       for (int i = 0;  i <= n; i++) {
            dp[I, k] = dp [I, k+1];
            If (I-  k >= 0)
                 dp [i,k] = dp [i,k] + dp [i-k,k];
        }
    }
   return dp [n, m];
}