We were discussing how to compare two models. For example, the number of data points must be larger than the number of parameters. if we increase the number of parameters, it will result in goodness of fit and better Chi-square. Consequently the model with higher number of parameters does better.This is articulated as a metric called the mean squared error which is chi-squared divided by degrees of freedom. MSE uses them as numerator and denominator so it represents a trade-off between over-parameterized model and a lower chi-square. A model with a fewer parameters but lower chi-square is preferred.
We will review 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 hte chi-square then the distribution of difference in subsequent parameters from a0 should be similar to that of the corresponding calculation with true parameters. #codingexercise
Find the Lobb Number. This counts the number of ways n+m open parantheses can be arranged to form the start of a valid sequence of balanced parantheses.
double GetLobb(double n, double m)
{
return ((2 * m + 1)  * GetNChooseK(2 * n , m + n)) / (m+n+1);
}
double GetNChooseK(double n, double k)

{

if (k <0 || k > n) return 0;

if (n < 0) return 0;

return Factorial(n) / (Factorial(n-k) * Factorial(k));

}

double Factorial(double n)

{

 if (n <= 1) return 1;

 return n * Factorial(n-1);

}

double GetNChooseKDP(double n, double k)
{
if (n < 0 || k < 0) return 0;
if (k == 0 || k == n)
    return 1;
return GetNChooseKDP(n-1, k-1) + GetNChooseKDP(n-1,k);
}

The NChooseK is also the binomial coefficient.