Related papers: Benford's law: A theoretical explanation for base …
This article provides a brief overview on a range of basic dynamical systems that conform to the logarithmic distribution of significant digits known as Benford's law. As presented here, most theorems are special cases of known, more…
We describe our solution to the Banff 2 challenge problems as well as the outcomes.
As probabilistic systems gain popularity and are coming into wider use, the need for a mechanism that explains the system's findings and recommendations becomes more critical. The system will also need a mechanism for ordering competing…
Claims of exceptions to the second law of thermodynamics are generally met with extreme skepticism that is quite reasonable given the great confidence placed in the second law. But what specifically is the basis for that confidence? The…
We prove that many sequences of positive numbers $(a_n)$ defined by finite linear difference equations $a_{n+k}=c_{k-1}a_{n+k-1}+...+c_0a_n$ with suitable non negative reals coefficients $c_i$ satisfy Bendford's Law on the first digit in…
The theory of bi-Hamiltonian systems has its roots in what is commonly referred to as the "Lenard recursion formula". The story about the discovery of the formula told by Andrew Lenard is the subject of this article.
A statistical model for the fragmentation of a conserved quantity is analyzed, using the principle of maximum entropy and the theory of partitions. Upper and lower bounds for the restricted partitioning problem are derived and applied to…
There is wide support in logic, philosophy, and psychology for the hypothesis that the probability of the indicative conditional of natural language, $P(\textit{if } A \textit{ then } B)$, is the conditional probability of $B$ given $A$,…
We study a generalisation of Simpson reversal (also known as Simpson's paradox or the Yule-Simpson effect) to $2 \times 2 \times 2$ contingency tables and characterise the cases for which it can and cannot occur with two…
The paper gives a simple algebraic description, and background justification, for the Bowley Ratio, the relative returns to labour and capital, in a simple economy.
In this paper, we intend to revisit Theorem 2 of [3] formulating it in a way that, weakening the hypotheses and, at the same time, highlighting the richer conclusion allowed by the proof, it can potentially be applicable to a broader range…
The main purpose of this work is to introduce and analyse some generalizations of diverse superposition rules for first-order differential equations to the setting of second-order differential equations. As a result, we find a way to apply…
Many legal cases require decisions about causality, responsibility or blame, and these may be based on statistical data. However, causal inferences from such data are beset by subtle conceptual and practical difficulties, and in general it…
We present a definition of cause and effect in terms of decision-theoretic primitives and thereby provide a principled foundation for causal reasoning. Our definition departs from the traditional view of causation in that causal assertions…
The purpose of this paper is to prove directly, by an elementary method, the Poisson probability law. This proof is offered as an alternative to the more usual derivation from binomial distribution in the limit of small probabilities. The…
We give some results and conjectures about recurrence relations for certain sequences of binomial sums.
Feller's classic text 'An Introduction to Probability Theory and its Applications' contains a derivation of the well known significant-digit law called Benford's law. More specifically, Feller gives a sufficient condition ("large spread")…
In this paper, we present betting strategy of a football game using probability theory. We know all betting houses offer slightly unfair odds towards the player. Here we discuss a simple way to figure out which betting house is offering…
For arbitrary Borel probability measures with compact support on the real line, characterizations are established of the best finitely supported approximations, relative to three familiar probability metrics (Levy, Kantorovich, and…
We consider an extension of first-order logic with a recursion operator that corresponds to allowing formulas to refer to themselves. We investigate the obtained language under two different systems of semantics, thereby obtaining two…