Related papers: A Novel Approach to Probability
The derivation of the maximum entropy distribution of particles in boxes yields two kinds of distributions: a "bell-like" distribution and a long-tail distribution. The first one is obtained when the ratio between particles and boxes is…
The Boltzmann distribution (the most probable distribution) is one of the most important concepts used in physics, chemistry and biology. Suppose we put the system initially in one of the less probable state then the system will find the…
Let $X_1, \ldots, X_n$ be independent random points drawn from an absolutely continuous probability measure with density $f$ in $\mathbb{R}^d$. Under mild conditions on $f$, we derive a Poisson limit theorem for the number of large…
There are two fair ways to distribute particles in boxes. The first way is to divide the particles equally between the boxes. The second way, which is calculated here, is to score fairly the particles between the boxes. The obtained power…
This is an attempt to address diffusion phenomena from the point of view of information theory. We imagine a regular hamiltonian system under the random perturbation of thermal (molecular) noise and chaotic instability. The irregularity of…
Given an arbitrary continuous probability density function, it is introduced a conjugated probability density, which is defined through the Shannon information associated with its cumulative distribution function. These new densities are…
Randomness is viewed through an analogy between a physical quantity, density of gas, and a mathematical construct -- probability density. Boltzmann's deduction of equilibrium distribution of ideal gas placed in an external potential field…
Commonly observed patterns typically follow a few distinct families of probability distributions. Over one hundred years ago, Karl Pearson provided a systematic derivation and classification of the common continuous distributions. His…
We comment on, elaborate, and extend the work of Warren Ewens and Herbert Wilf, described in their http://www.pnas.org/content/104/27/11189.full.pdf about the maximum in balls-and-boxes problem. In particular we meta-apply their ingenious…
A discrete-time stochastic process derived from a model of basketball is used to generalize any discrete distribution. The generalized distributions can have one or two more parameters than the parent distribution. Those derived from…
We find the laws for the spreading of the spatial widths (parallel and transverse to the direction of average motion) of the relativistic position probability density for a massive, spinless particle. We find that when the momentum width of…
A numerical experiment of ideal stochastic motion of a particle subject to conservative forces and Gaussian noise reveals that the path probability depends exponentially on action. This distribution implies a fundamental principle…
Zipf's law is the most common statistical distribution displaying scaling behavior. Cities, populations or firms are just examples of this seemingly universal law. Although many different models have been proposed, no general theoretical…
The Bernoulli sieve is the infinite "balls-in-boxes" occupancy scheme with random frequencies $P_k=W_1...W_{k-1}(1-W_k)$, where $(W_k)_{k\in\mn}$ are independent copies of a random variable $W$ taking values in $(0,1)$. Assuming that the…
Science in the 21st century seems to be governed by novel approaches involving interdisciplinary work, systemic perspectives and complexity theory concepts. These new paradigms force us to leave aside our elder mechanistic approaches and…
We study the long-time behavior of the probability density associated with the decoupled continuous-time random walk which is characterized by a superheavy-tailed distribution of waiting times. It is shown that if the random walk is…
In the last years, researchers have realized the difficulties of fitting power-law distributions properly. These difficulties are higher in Zipf's systems, due to the discreteness of the variables and to the existence of two representations…
We consider in this paper an urn and ball problem with replacement, where balls are with different colors and are drawn uniformly from a unique urn. The numbers of balls with a given color are i.i.d. random variables with a heavy tailed…
We propose a new method for the calculation of the statistical properties, as e.g. the entropy, of unknown generators of symbolic sequences. The probability distribution $p(k)$ of the elements $k$ of a population can be approximated by the…
General equilibrium equations in economics play the same role with many-body Newtonian equations in physics. Accordingly, each solution of the general equilibrium equations can be regarded as a possible microstate of the economic system.…