Another day, another blog. Today in my very interactive Astro-statistics-Python-coding class, I gained a better understanding about the concepts of probability density/distribution function and the cumulative probability distribution function. I will explain how the other interns and I have made significantly more progress in enhancing our Python scripts to do much more analysis regarding our use of simulated data that constitutes a linear trend with random noise added to it. If you want to follow along, here is my code that you can copy and paste into a script or go to RAW and then download (with the "wget" Terminal command). The next series of plots are but one of the many possible outcomes from executing my code:

>>>execfile('stat_ex_uncertainties')

>>>cdf(a_vals, plot=True) # this can be ran however many times you wish

A quick aside: why does the contour plot have that slope to it?

If you run my "cdf(a_vals, plot=True)" function multiple times, the values and eccentricities of the regions may change but it will still have that tilt. As I am writing this blog, I force myself to explain these details meticulously, which in turn compels me to consider all questions that might arise in the thoughts of my audience. The question I have posed is one I am still contemplating. As of now, I think that the slope of the ellipses attests to how well the two parameters are being constrained. The linear slope parameter \(a_1\) is being constrained more than the intercept \(a_0\). I am inclined to believe that this is why the regions are stretched horizontally more than they are stretched vertically.

The next two plots reiterate and confirm what the contour plot is showing but from a 1-dimensional point of view. (Thus the need for 2 plots to express the results of the 2D contour.)

Now that you're all caught up to speed, I can discuss the extra steps (since my last post) we have done for this exercise. I guess you can call this

**Phase III**just to stay consistent with the previous couple posts.
Figures 3 and 4 illustrate the Probability Density Function (PDF) that we know to be a Gaussian. This means that the contour plot of Figure 2 yields a Gaussian distribution as well. The data in those plots also show the best, or most satisfactory, value for each fitting parameter. However, one of those circular data points may not reveal the most satisfactory value for the fitting parameter despite the fact that one of those points truly does look like it is at the peak of the curve (in both plots). To find a value that is truly at the peak of the curve, I had to first integrate the PDF produced by my contour plot in Figure 2 and subsequently interpolate the resultant curve in order to find the x-value that corresponds to where the area under the curves of Figures 3 and 4 equate to \(\frac{1}{2} \) the total area.

*to be continued...*

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