We have here a paradox: what theory is it that is not correct unless it leaves open the possibility that it may be incorrect? Answer: the theory of probability.

P.W. Bridgman, The Nature of the Physical Theory

While reasoning on one of the biological problems I have been spending most of my time on, I encounterd a random variable that was the ratio between two random variables with a Beta distribution. Let’s say that X_1\,\sim\,B(\alpha_1,\beta_1) and X_2\,\sim\,B(\alpha_2,\beta_2). Then, if we define the random variable Y\,=\,\frac{X_1}{X_2}, what is its density? What about its cumulative distribution function? And what can we say about its moments? I answered all these questions in this paper. The density, in particualr, is given by:

On the exact same subject there is a 1999 paper I am currently comparing to mine (Pham-Gia 1999). While writing my own paper I did not check for other works on the same topic, but if I had had to bet I would have said that this calculation had been performed a century ago; so I was astonished when I discovered that this density was first published only 20 years ago! This is further proof of how recent the application of calculus to statistics is.

PS. I have just checked the density by Thu Pham-Gia: it is equivalent to the one I found, even though they appear different at a first glance. I have added a paragraph to my paper to show the equivalence between these two densities. It is also worth mentioning that in addition, I calculated the cumulative distribution function, the expectation, the second-order moment, and the variance, while Pham-Gia presented the density along with related subjects and applications, in his 1999 paper.


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