Suppose that a box contains \( N \) balls, of which \( \textcolor{red}{M} \) are red ("success") and \( \textcolor{red}{N-M} \) are of another color ("failure"). We now draw balls with replacement (notably, if we draw without replacement, we would end up with hypergeometric distribution), so that after each draw the selected ball is returned to the box before the next draw. Consequently, each draw is independent, and the probability of success remains constant,

\begin{equation} p=\frac{M}{N}. \end{equation}

Let \( Y \) denote the number of red balls drawn ("successes") in the \( n \) draws ("experiments"). We wish to determine the probability that exactly \( \textcolor{red}{k} \) of the \( n \) (all) draws are successes.

In quite a few statistical physics books, probability density functions (abbr. PDF) are introduced instantaneously by using the Dirac delta function. In my view, this approach is not very intuitive. While reading Heinz's book on mathematical modeling [1], I came across a more intuitive explanation, and I would like to share it here on Physics of Risk.

During the summer, I received an email from a researcher at university abroad inquiring about how we (myself and my senior colleague, prof. B. Kaulakys) derived the approximation for the characteristic function of bounded Pareto distribution (for the particular case with \( \alpha = 1 \)). The approximation was given in [1], but derivation turned out to be somewhat more involved than I remembered. Originally, I derived it with the help of Wolfram Engine, while my colleague consulted various mathematical formula compendiums. Unfortunately, I was unable to find any notes containing the derivation. So I have attempted to reconstruct a combined approach from scratch. You'll find the derivation below.

So, we are back for a new academic year! Over the summer, our group continued reading books on statistics. Our summer readings, both old and new, has inspired a few posts that will be published on the blog soon. We hope you will find them enlightening!

Oh, and by the way, this September is our 21st September since we started Physics of Risk blog (if we include the 2006–2010 period when the posts were written in Lithuanian). 20 years have gone by!

Do you have plans for summer holidays? If not, here is a quick idea presented to you by Woolly Benguin at MathsJam 2023.