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Related Concept Videos

Probability Distributions01:32

Probability Distributions

The probability of a random variable x  is the likelihood of its occurrence. A probability distribution represents the probabilities of a random variable using a formula, graph, or table. There are two types of probability distribution– discrete probability distribution and continuous probability distribution.
A discrete probability distribution is a probability distribution of discrete random variables. It can be categorized into binomial probability distribution and Poisson probability...
Applications of Integration to Probability Density Functions01:27

Applications of Integration to Probability Density Functions

Continuous probability distributions are used to model random variables that can take on any real value within a specified range. These variables do not take on isolated or countable values but rather exist on a continuum. For example, the height of an individual can be measured with increasing precision—such as 163.5 or 165.25 centimeters—demonstrating that height is a continuous random variable.The behavior of such variables is described using a probability density function (PDF), which...
Probability Histograms01:17

Probability Histograms

A probability histogram is a visual representation of a probability distribution. Similar a typical histogram, the probability histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents. The vertical axis is labeled with probability. Each rectangular bar in the histogram is 1 unit wide, which suggests that the area under each bar equals the probability, P(x), where x is 1, 2, 3, and so on.
Probability in Statistics01:14

Probability in Statistics

Probability is the likelihood of an event occurring. The term event is defined as a collection of results of a procedure. An event is a simple event when an outcome cannot be divided into simpler parts.
An example of a simple event is a coin toss. The result of a coin toss is either a head or a tail. Here, head and tail are two simple events. These two simple events make up the sample space. Further, the probability of an event occurring falls within the range of 0 to 1. The probability of an...
Superposition Theorem01:18

Superposition Theorem

The superposition principle is a fundamental concept stating that in a linear circuit, the voltage across (or current through) an element can be determined by summing the individual contributions of each independent source acting in isolation. When dealing with linear circuits containing multiple independent sources, this principle serves as a valuable tool for analysis. To apply the superposition principle effectively, one should focus on a single independent source at a time while...
Interference and Superposition of Waves01:07

Interference and Superposition of Waves

When two waves of the same nature occur in the same region simultaneously, they result in interference. Interference of waves implies that the net effect of the waves is the sum of the individual waves' effects. However, it does not imply that the individual waves affect the propagation of other waves.
Interference occurs in mechanical waves, such as sound waves, waves on a string, and surface water waves. Mechanical waves correspond to the physical displacement of particles. Hence,...

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A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
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Superpositions of probability distributions.

Petr Jizba1, Hagen Kleinert

  • 1ITP, Freie Universität Berlin, Arnimallee 14 D-14195 Berlin, Germany. jizba@physik.fu-berlin.de

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 15, 2008
PubMed
Summary

Superpositions of Gaussian distributions, crucial in quantum theory and finance, can introduce memory effects. This study reveals smearing distributions that preserve the semigroup property, simplifying complex calculations and extending to quantum mechanics.

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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Generation and Coherent Control of Pulsed Quantum Frequency Combs

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Area of Science:

  • Probability theory
  • Quantum mechanics
  • Financial mathematics

Background:

  • Superpositions of Gaussian distributions are fundamental in quantum theory and financial markets.
  • These superpositions may violate the Chapman-Kolmogorov semigroup relation due to memory effects.

Purpose of the Study:

  • Derive the general form of smearing distributions in variance (v) that preserve the semigroup property.
  • Simplify Kramers-Moyal equations and path integral calculus for these distributions.
  • Extend the smearing technique to quantum mechanics.

Main Methods:

  • Mathematical derivation of smearing distributions.
  • Application of the smearing technique to Kramers-Moyal equations.
  • Implementation within path integral calculus.

Main Results:

  • Identified smearing distributions that maintain the semigroup property for Gaussian superpositions.
  • Demonstrated simplification of Kramers-Moyal equations for conditional probabilities.
  • Showcased easier evaluation of path integrals compared to initial integrals.
  • Extended the methodology to quantum mechanical systems.

Conclusions:

  • The developed smearing technique effectively handles memory effects in Gaussian superpositions.
  • This approach offers significant computational advantages in probability theory and quantum mechanics.
  • The method provides a unified framework for analyzing complex systems in both finance and physics.