6,828 research outputs found
Expressions for the entropy of binomial-type distributions
We develop a general method for computing logarithmic and log-gamma expectations of distributions. As a result, we derive series expansions and integral representations of the entropy for several fundamental distributions, including the Poisson, binomial, beta-binomial, negative binomial, and hypergeometric distributions. Our results also establish connections between the entropy functions and to the Riemann zeta function and its generalizations
A simple derivation and classification of common probability distributions based on information symmetry and measurement scale
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 approach was phenomenological: a differential equation that
generated common distributions without any underlying conceptual basis for why
common distributions have particular forms and what explains the familial
relations. Pearson's system and its descendants remain the most popular
systematic classification of probability distributions. Here, we unify the
disparate forms of common distributions into a single system based on two
meaningful and justifiable propositions. First, distributions follow maximum
entropy subject to constraints, where maximum entropy is equivalent to minimum
information. Second, different problems associate magnitude to information in
different ways, an association we describe in terms of the relation between
information invariance and measurement scale. Our framework relates the
different continuous probability distributions through the variations in
measurement scale that change each family of maximum entropy distributions into
a distinct family.Comment: 17 pages, 0 figure
Statistical Models of Nuclear Fragmentation
A method is presented that allows exact calculations of fragment multiplicity
distributions for a canonical ensemble of non-interacting clusters.
Fragmentation properties are shown to depend on only a few parameters.
Fragments are shown to be copiously produced above the transition temperature.
At this transition temperature, the calculated multiplicity distributions
broaden and become strongly super-Poissonian. This behavior is compared to
predictions from a percolation model. A corresponding microcanonical formalism
is also presented.Comment: 12 pages, 5 figure
How multiplicity determines entropy and the derivation of the maximum entropy principle for complex systems
The maximum entropy principle (MEP) is a method for obtaining the most likely
distribution functions of observables from statistical systems, by maximizing
entropy under constraints. The MEP has found hundreds of applications in
ergodic and Markovian systems in statistical mechanics, information theory, and
statistics. For several decades there exists an ongoing controversy whether the
notion of the maximum entropy principle can be extended in a meaningful way to
non-extensive, non-ergodic, and complex statistical systems and processes. In
this paper we start by reviewing how Boltzmann-Gibbs-Shannon entropy is related
to multiplicities of independent random processes. We then show how the
relaxation of independence naturally leads to the most general entropies that
are compatible with the first three Shannon-Khinchin axioms, the
(c,d)-entropies. We demonstrate that the MEP is a perfectly consistent concept
for non-ergodic and complex statistical systems if their relative entropy can
be factored into a generalized multiplicity and a constraint term. The problem
of finding such a factorization reduces to finding an appropriate
representation of relative entropy in a linear basis. In a particular example
we show that path-dependent random processes with memory naturally require
specific generalized entropies. The example is the first exact derivation of a
generalized entropy from the microscopic properties of a path-dependent random
process.Comment: 6 pages, 1 figure. To appear in PNA
Generalized Transmuted Family of Distributions: Properties and Applications
We introduce and study general mathematical properties of a new generator of continuous distributions with two extra parameters called the Generalized Transmuted Family of Distributions. We investigate the shapes and present some special models. The new density function can be expressed as a linear combination of exponentiated densities in terms of the same baseline distribution. We obtain explicit expressions for the ordinary and incomplete moments and generating function, Bonferroni and Lorenz curves, asymptotic distribution of the extreme values, Shannon and R´enyi entropies and order statistics, which hold for any baseline model. Further, we introduce a bivariate extension of the new family. We discuss the different methods of estimation of the model parameters and illustrate the potential application of the model via real data. A brief simulation for evaluating Maximum likelihood estimator is done. Finally certain characterziations of our model are presented
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