Astro-Stats & Python : Recap

In my Grad level Intro to Astro-Stats for Undergraduates  post and its sequel, I explained the purpose, meaning and results of one exercise assigned to me within the Banneker Institute.  Recently, we did the same exercise but by using a different method--Monte Carlo Bootstrap.

To quickly recap on the previous posts regarding this Astro-Stats coursework:

• I wrote code that could forge a polynomial equation solely based on the number of coefficients you feed the code.  For example, if I give it the numbers 5, 6 and 100, then my codes knows that the highest degree for this polynomial is 2 and therefore the equation is \(y = 5 + 6x + 100x^2\).  For another example, I can give it the numbers 0, 250.3, -2.76 and 20 and my code will use the following equation in its calculations: \(y = 0 + 250.3x - 2.76x^2 + 20x^3 \).
• When the various "x" values and coefficients are given, the code will simulate realistic measured data with a random number generator (that follows a normal distribution).  Therefore my equation used to simulate the data would be \(y = 0 + 250.3x - 2.76x^2 + 20x^3 +\) random_number
  
  
  
• In my previous posts, I used an example where only 2 coefficients were chosen \(a_0\) and \(a_1\).  The goal is to approximate the value of those coefficients by using the simulated data (that has noise in it).  In order to do this, I need to make initial guesses as to what those true values are.  I made ~100 guesses for both coefficients.
  
  
• I used \(\chi^2\) statistics on the simulated data and the 100+ coefficient guesses in order to find the Likelihood of every possible combination those 2 sets of guesses.  
  
  
• With those Likelihoods stored inside of a matrix, I found the Cumulative Distribution Function of the Likelihoods and subsequently used interpolation to find the value for the coefficient  that was at the peak of the Probability Distribution Function.  This was done for both coefficients.
  
  
• Lastly, I found the uncertainty of my newly approximated coefficients.

I'll show the Monte Carlo Bootstrap exercise later.