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Applications of Normal Distribution01:22

Applications of Normal Distribution

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The normal distribution is a useful statistical tool. One of its practical applications is determining the door height after considering the normal distribution of heights of persons, such that many can pass through it easily without striking their heads. The normal distribution can also determine the probability of a person having a height less than a specific height.
The heights of 15 to 18-year-old males from Chile from 1984 to 1985 followed a normal distribution. The mean height is 172.36...
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The uniform distribution is a continuous probability distribution of events with an equal probability of occurrence. This distribution is rectangular.
Two essential properties of this distribution are
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In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
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Normal Distribution01:11

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The normal, a continuous distribution, is the most important of all the distributions. Its graph is a bell-shaped symmetrical curve, which is observed in almost all disciplines. Some of these include psychology, business, economics, the sciences, nursing, and, of course, mathematics. Some instructors may use the normal distribution to help determine students’ grades. Most IQ scores are normally distributed. Often real-estate prices fit a normal distribution. The normal distribution is...
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Probability Distributions01:32

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 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.
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Random Variables01:09

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A random variable is a single numerical value that indicates the outcome of a procedure. The concept of random variables is fundamental to the probability theory and was introduced by a Russian mathematician, Pafnuty Chebyshev, in the mid-nineteenth century.
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Large Scale Energy Efficient Sensor Network Routing Using a Quantum Processor Unit
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Entropy-Regularized Optimal Transport on Multivariate Normal and q-normal Distributions.

Qijun Tong1, Kei Kobayashi1

  • 1Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama 223-8522, Japan.

Entropy (Basel, Switzerland)
|April 3, 2021
PubMed
Summary

Entropy-regularized optimal transport provides an efficient approximation for Wasserstein distance. This study derives explicit forms for this cost on normal and q-normal distributions, offering theoretical insights into entropy regularization effects.

Keywords:
Tsallis entropyWasserstein distanceentropy regularizationoptimal transportationq-normal distribution

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Area of Science:

  • Statistics
  • Machine Learning
  • Optimal Transport Theory

Background:

  • Probability measures and their distances are fundamental in statistics and machine learning.
  • Wasserstein distance is crucial but computationally expensive to calculate.
  • Entropy regularization offers an efficient approximation for Wasserstein distance.

Purpose of the Study:

  • To theoretically analyze entropy-regularized optimal transport.
  • To derive explicit forms for entropy-regularized optimal transport costs.
  • To understand the impact of entropy regularization on statistical measures.

Main Methods:

  • Focus on entropy-regularized optimal transport for multivariate normal and q-normal distributions.
  • Derivation of explicit formulas for the optimal transport cost.
  • Development of an entropy-regularized Kantorovich estimator.

Main Results:

  • Obtained explicit forms for entropy-regularized optimal transport costs on normal and q-normal distributions.
  • Provided theoretical understanding of entropy regularization's effect, previously empirical.
  • Derived a novel entropy-regularized Kantorovich estimator.

Conclusions:

  • The explicit forms offer a theoretical basis for understanding entropy regularization in optimal transport.
  • The derived Kantorovich estimator is a novel contribution for probability measure estimation.
  • These findings are foundational for generalizing entropy-regularized optimal transport.