After having spent a few evenings exploring Python’s various math & science modules, specifically those dealing with probability, simulation and statistics, perhaps it’s time for a brief summary.
In my previous recent posts, I’ve gone thru concepts such as mild & wild randomness, and shown how they relate to different probability distributions.For this, I’ve used the Blindfolded Archer, and the Drunken Walk (or Random Walk) as metaphors.
Below a few graphs to illustrate these concepts. First, a graph showing the results of four different drunken walks, each based on a different probability distribution. The red spots illustrate the starting and ending points for each of these random walks, where each new step is randomly chosen, relative to current position. Since the scales of each of the four graphs are different, it’s easy to get an erroneous picture of what’s going on.
From the above graph, if you don’t pay attention to the different scales, it looks like that total distance covered is more or less the same, but looking at the scales you realize that’s not the case at all.
Therefore, let’s make the scale the same for all four walks:
Here we can see the ‘outliers’, the big steps, present in the Cauchy and Levy distributions, while the uniform drunken walk is very cohesive, and the normal drunken walk is only slightly more adventurous.
As an aside, an interesting observation about the Levy stable distribution is that it can be made gaussian by chosing the appropriate α and β parameters, 2 & 0, respectively. With other parameters, the Levy distribution can look very much like the Cauchy.
Another illustration of the impact of the various distributions can be gained from my previous example of the skilled vs blindfolded archers: the below graph illustrates the performance of archers whose skills are modelled by the four different probability distributions: Normal/Gaussian, Uniform, Cauchy & Levy stable:
As can be seen, the archer with normal distribution produces very consistent results, thus demonstrating mild randomness (and a steady eye and hand!) while the archer with Cauchy distribution spreads his arrows all over the place. The uniform archer spreads his shots equally frequently within a firm range, and lastly, the Levy archer, depending on parameters, can be made to execute as being wild or mild.