Probability Distributions of Continuous Variables

2003 ◽  
pp. 141-156
Author(s):  
W.J. DeCoursey
Keyword(s):  
Probability Distributions ◽  
Continuous Variables

Continuous Random Numbers

2019 ◽  
pp. 54-69
Author(s):  
Robert H. Swendsen
Keyword(s):  
Probability Distributions ◽  
Random Variables ◽  
Delta Function ◽  
Ideal Gas ◽  
Dirac Delta Function ◽  
Bayesian Probability ◽  
Random Numbers ◽  
Continuous Variables ◽  
Dirac Delta ◽  
Discrete Random Variables

The theory of probability developed in Chapter 3 for discrete random variables is extended to probability distributions, in order to treat the continuous momentum variables. The Dirac delta function is introduced as a convenient tool to transform continuous random variables, in analogy with the use of the Kronecker delta for discrete random variables. The properties of the Dirac delta function that are needed in statistical mechanics are presented and explained. The addition of two continuous random numbers is given as a simple example. An application of Bayesian probability is given to illustrate its significance. However, the components of the momenta of the particles in an ideal gas are continuous variables.

scholarly journals SU(2) Symmetry of Qubit States and Heisenberg–Weyl Symmetry of Systems with Continuous Variables in the Probability Representation of Quantum Mechanics

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1099 ◽  
Author(s):  
Peter Adam ◽  
Vladimir A. Andreev ◽  
Margarita A. Man’ko ◽  
Vladimir I. Man’ko ◽  
Matyas Mechler
Keyword(s):  
Probability Distributions ◽  
Star Product ◽  
Kinetic Equations ◽  
Random Variables ◽  
Continuous Variables ◽  
Probability Representation ◽  
Density Operators ◽  
Symmetry Properties ◽  
Weyl Symmetry ◽  
Qubit States

In view of the probabilistic quantizer–dequantizer operators introduced, the qubit states (spin-1/2 particle states, two-level atom states) realizing the irreducible representation of the S U ( 2 ) symmetry group are identified with probability distributions (including the conditional ones) of classical-like dichotomic random variables. The dichotomic random variables are spin-1/2 particle projections m = ± 1 / 2 onto three perpendicular directions in the space. The invertible maps of qubit density operators onto fair probability distributions are constructed. In the suggested probability representation of quantum states, the Schrödinger and von Neumann equations for the state vectors and density operators are presented in explicit forms of the linear classical-like kinetic equations for the probability distributions of random variables. The star-product and quantizer–dequantizer formalisms are used to study the qubit properties; such formalisms are discussed for photon tomographic probability distribution and its correspondence to the Heisenberg–Weyl symmetry properties.

scholarly journals Updating Short-Term Probabilistic Weather Forecasts of Continuous Variables Using Recent Observations

2011 ◽  
Vol 26 (4) ◽  
pp. 564-571 ◽  
Author(s):  
Thomas N. Nipen ◽  
Greg West ◽  
Roland B. Stull
Keyword(s):  
Recent Observation ◽  
Probability Distributions ◽  
Computational Cost ◽  
Absolute Error ◽  
Transition Function ◽  
Cumulative Distribution ◽  
Continuous Variables ◽  
Probabilistic Forecasts ◽  
Forecasting System ◽  
Highly Correlated

Abstract A statistical postprocessing method for improving probabilistic forecasts of continuous weather variables, given recent observations, is presented. The method updates an existing probabilistic forecast by incorporating observations reported in the intermediary time since model initialization. As such, this method provides updated short-range probabilistic forecasts at an extremely low computational cost. The method models the time sequence of cumulative distribution function (CDF) values corresponding to the observation as a first-order Markov process. Verifying CDF values are highly correlated in time, and their changes in time are modeled probabilistically by a transition function. The effect of the method is that the spread of the probabilistic forecasts for the first few hours after an observation has been made is considerably narrower than the original forecast. The updated probability distributions widen back toward the original forecast for forecast times far in the future as the effect of the recent observation diminishes. The method is tested on probabilistic forecasts produced by an operational ensemble forecasting system. The method improves the ignorance score and the continuous ranked probability score of the probabilistic forecasts significantly for the first few hours after an observation has been made. The mean absolute error of the median of the probability distribution is also shown to be improved.

scholarly journals An information integration account for causal generalizations in uncertain causal networks

2022 ◽  
Author(s):  
Moyun Wang
Keyword(s):  
Information Integration ◽  
Response Pattern ◽  
Probability Distributions ◽  
Joint Effect ◽  
Continuous Variables ◽  
Conditional Probabilities ◽  
Joint Function ◽  
Common Cause ◽  
Causal Networks ◽  
Collateral Effects

In reasoning about common cause networks, given that a cause generates an effect, people often need to infer how likely the cause generate another effect. This causal generalization question has not systematically been investigated in previous research. We propose the information integration account for causal generalizations in uncertain casual networks with dichotomized continuous variables. It predicts that causal generalization is the joint function of conditional probabilities of causal links and cause strength indicated by the proportion of present collateral effects. Two experiments investigated causal generalizations in uncertain causal networks with and without probability distributions, respectively. It was found that in the presence of probability distributions there was the joint effect of conditional probability and cause strength on causal generalization; in the absence of probability distributions causal generalization depend only on cause strength. The overall response pattern favors the information integration account over the other alternative accounts.

scholarly journals Probability Representation of Quantum States

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 549
Author(s):  
Olga V. Man’ko ◽  
Vladimir I. Man’ko
Keyword(s):  
Quantum Mechanics ◽  
Open Systems ◽  
Probability Distributions ◽  
Free Particle ◽  
Classical Mechanics ◽  
Quantum Tomography ◽  
Quantum States ◽  
Space Representation ◽  
Continuous Variables ◽  
Probability Representation

The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born’s rule and recently suggested method of dequantizer–quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical–like equations for the probability distributions determining the quantum system states. Relations to phase–space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.

Application of Bayesian Networks in Multi-Hazard Safety Assessment of Nuclear Power Plants

2020 ◽  
Author(s):  
Varenya Kumar D. Mohan ◽  
Philip Vardon ◽  
James Daniell ◽  
Pierre Gehl ◽  
Andreas Schafer ◽  
...  
Keyword(s):  
Risk Assessment ◽  
Bayesian Networks ◽  
Bayesian Network ◽  
Nuclear Power ◽  
Power Plants ◽  
Nuclear Power Plants ◽  
Probability Distributions ◽  
Sensitivity Analyses ◽  
Continuous Variables ◽  
Safety Assessments

<p>Low probability events occurring in sequence, within a limited operational time (damage and recovery window between events), are a key consideration in multi-hazard safety assessments of nuclear power plants (NPPs). Cascading effects from hazards and associated event sequences could potentially have a significant impact on risk estimates. The Bayesian network can act as a framework to consider aforementioned statistical dependencies between various hazards in multi-risk analyses of nuclear power plants.</p><p>Within the EU project NARSIS (New Approach to Reactor Safety Improvements), a Bayesian network-based risk assessment framework was developed to perform multi-hazard risk assessment of NPPs.</p><p>The Bayesian network method was applied for an external-event related station blackout (SBO) scenario at a NPP. Earthquake, flooding, and tornado were among the hazards considered at a decommissioned NPP site location in Europe. Both hazard dependency in time as well as a cascading scenario was also considered. The hazards, their interactions and the fragilities of selected systems, structures and components within the nuclear power plant were represented in the network and their probability distributions were obtained based on the multi-hazard and fragility approaches adopted within the NARSIS project.</p><p>Sensitivity analyses in the network were used to identify key hazards and interactions. Most influential hazard combinations and ranges of intensity measures were identified through diagnostic inference in the network. Discretisation of continuous variables (hazard curves in this case) is a key aspect of performing inference in Bayesian networks. The effect of various levels of discretisation of hazard probability distributions was assessed, to identify suitable discretisations of hazard data.</p><p>This application demonstrates the use and advantages of the Bayesian network methodology, developed in the NARSIS project, for probabilistic safety assessments of NPPs.</p>

scholarly journals Probability Representation of Quantum Mechanics Where System States Are Identified with Probability Distributions

2020 ◽  
Vol 2 (1) ◽  
pp. 64-79 ◽  
Author(s):  
Vladimir Chernega ◽  
Olga Man'ko ◽  
Vladimir Man'ko
Keyword(s):  
Quantum Mechanics ◽  
Probability Distributions ◽  
Random Variables ◽  
Time Dependent ◽  
Continuous Variables ◽  
Probability Representation ◽  
Classical Statistics ◽  
Quantum Observables ◽  
System States ◽  
Dependent Probability

The probability representation of quantum mechanics where the system states are identified with fair probability distributions is reviewed for systems with continuous variables (the example of the oscillator) and discrete variables (the example of the qubit). The relation for the evolution of the probability distributions which determine quantum states with the Feynman path integral is found. The time-dependent phase of the wave function is related to the time-dependent probability distribution which determines the density matrix. The formal classical-like random variables associated with quantum observables for qubit systems are considered, and the connection of the statistics of the quantum observables with the classical statistics of the random variables is discussed.

Eliciting subjective probability distributions on continuous variables

1978 ◽  
Vol 21 (3) ◽  
pp. 379-391 ◽  
Author(s):  
David A. Seaver ◽  
Detlof von Winterfeldt ◽  
Ward Edwards
Keyword(s):  
Subjective Probability ◽  
Probability Distributions ◽  
Continuous Variables
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