In the 1738, Daniel Bernoulli, the very same known for his contribution to fluid dynamics, in his paper in the "Commentaries of the Imperial Academy of Science of Saint Petersburg" described an interesting paradox. Let us assume that we have a fair 50-50 game in which the host tosses a coin until the tail appears. After each toss he pays a player \( 2^n \) (where \( n \) is a number of the toss) of money. The problem in question is - what is an optimal price for the game? Namely how much money the host should ask from a player, that he would be still motivated to play the game, yet also preventing unnecessary losses by the host.

The quantitative comparison of economic growth of various states is still an ambiguous task. Economists and statisticians use various estimates of Gross Domestic Product (GDP) taking into account inflation, population, exchange rates etc. Here we present a graphical comparison of GDP growth of various states aimed at the estimation of relative input of various states into regions or the world economy and at the measure of economic convergence. We choose the estimate of GDP in common currency US dollars calculated in current prices and current exchange rates. In order to compare different size states we use GDP normalized per capita. Such data is available at the World Bank Database.

Last time we have written on the power spectral density and we have "analyzed" deterministic periodic time series. This time we will consider spectral densities of some stochastic processes.

Here, on the Physics of Risk, we frequently talk about two essential statistical features of the time series - probability and spectral densities. The probability density function should well known to our readers - it is related to the distribution of time series values. So let us now discuss the power spectral density.

In classical physics differential equations is the main tool to mathematically describe dynamical systems. Having obtained the mathematical descriptions of the system we should be able to predict the evolution of the system. It is noted that the evolution of the classical systems is pretty trivial - no matter what the initial condition is the system will "find" the stable state. Usually dissipative forces (such as frictions) are to be blamed for this. Though not all systems are so simple...