As we have already discussed, beta prime distribution arises from a nonlinear transformation of the voter model. Furthermore, recently we have been relying a lot on the said transformation [1, 2, 3]. In those papers, we have been using some results derived for the CIR process as well. Thus, another interesting thing, which my colleague Rytis Kazakevičius has noted, was that the beta prime distribution can be obtained from the ratio of two independent Gamma-distributed random values. Why it is interesting? The stationary distribution of the CIR process is the Gamma distribution!

Have you heard of the beta prime distribution before? Until recently, I hadn't either. My colleague, Rytis Kazakevičius, recently surprised me by pointing out that the distribution we have been repeatedly encountering in nonlinear transformations of the noisy voter model has actually a proper name. It is known as the beta prime distribution. And our history with this distribution, goes back much further.

Recently, Veritasium has posted a video on Newcomb's paradox. This paradox is based on a decision-making problem. Namely, you are presented with two boxes. Box A is transparent and contains a small amount of money, while Box B is opaque. Box B might contain a large amount of money or be empty. Its contents are decided ahead of time by a machine, which has perfect prediction record. If the machine predicts that you will take only Box B, it will put large amount of money inside it. Otherwise, it will keep Box B empty. So, will you take only Box B (and believe in predictor accuracy) or will you take both boxes (and believe in game theory)? For more details on the paradox and its solutions watch the video below.

After watching the video above I wasn't particularly fond of the paradox. Mostly because the paradox seemed to be artificially contrived. After watching minute physics (previously known as One Minute Physics) video I still feel the same way, but surprisingly this paradox prompts us to consider the nature of our own reality. I encourage you to watch it as well.

I especially liked the part about random 5 year old falsely achieving incredible accuracy.

Our group, along with a few students, has been reading statistics handbook and refreshing our understanding of the basic statistics. I was given to cover a chapter about the central limit theorem, which reminded me that I had already given a similar presentation while being PhD student myself. While diving into the topic, I have noticed a couple things, which are usually glanced over in a typical statistics handbook. In the final post of this series, let me put an emphasis on the defining property of any stable distribution.

Our group, along with a few students, has been reading statistics handbook and refreshing our understanding of the basic statistics. Some time ago, I was given to cover a chapter about the central limit theorem, which reminded me that I had already given a similar presentation while being PhD student myself. While diving into the topic, I have noticed a couple things, which are usually glanced over in a typical statistics handbook. Let me share them with you.

This time we explore infinite divisibility. Our previous mathematical explorations of the stable distributions topic have relied on this property, because it simplifies many of the analytical derivations. But there are distributions, which are infinitely divisible but not stable. This time let us take a look at sums of Gamma distributed random variates.