scholarly journals A Theory Of Errors in Quantum Measurement

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2439-2450
Author(s):  
S. G. Rajeev
Keyword(s):  
Quantum Theory ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Probability Distribution ◽  
Quantum Measurement ◽  
Probability Distributions ◽  
Matrix Elements ◽  
Random Errors ◽  
The Central Limit Theorem ◽  
The Matrix

It is common to model the random errors in a classical measurement by the normal (Gaussian) distribution, because of the central limit theorem. In the quantum theory, the analogous hypothesis is that the matrix elements of the error in an observable are distributed normally. We obtain the probability distribution this implies for the outcome of a measurement, exactly for the case of traceless 2 × 2 matrices and in the steepest descent approximation in general. Due to the phenomenon of 'level repulsion', the probability distributions obtained are quite different from the Gaussian.

scholarly journals Understanding the Central Limit Theorem the Easy Way: A Simulation Experiment

Proceedings ◽  
2018 ◽  
Vol 2 (21) ◽  
pp. 1322
Author(s):  
Monica E. Brussolo
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Electronic Device ◽  
Probability Distributions ◽  
Online Simulation ◽  
The Central Limit Theorem ◽  
Sampling Distributions ◽  
Initial Stage ◽  
Random Samples

Using a simulation approach, and with collaboration among peers, this paper is intended to improve the understanding of sampling distributions (SD) and the Central Limit Theorem (CLT) as the main concepts behind inferential statistics. By demonstrating with a hands-on approach how a simulated sampling distribution performs when the data used has different probability distributions, we expect to clarify the notion of the Central Limit Theorem, and the use of samples in the hypothesis testing process for populations. This paper will discuss an initial stage to create random samples from a given population (using Excel) with collaboration of the students, which has been tested in the classroom. Then, based on that experience, a second stage in which we created an online simulation, controlled by the professor, and in which the students will participate during class time using an electronic device connected to internet. Students will create simple random samples from a variety of probability distributions simulated online in a collaborative way. Once the samples are generated, the instructor will combine and summarize the resulting sample statistics using histograms and the results will be discussed with the students. The objective is to teach some of the central topics of introductory statistics, the Central Limit Theorem and sampling distributions with an interactive and engaging approach.

Turbulent velocity fluctuations need not be Gaussian

1998 ◽  
Vol 376 ◽  
pp. 139-147 ◽  
Author(s):  
JAVIER JIMÉNEZ
Keyword(s):  
Energy Spectrum ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Probability Distribution ◽  
Experimental Evidence ◽  
Turbulent Velocity ◽  
Spectral Slope ◽  
Velocity Fluctuations ◽  
The Central Limit Theorem ◽  
Coherence Lengths

It is noted that the central limit theorem does not constrain the probability distribution of the components of the turbulent velocity fluctuations to be Gaussian, if their spectral slope is steeper than k−1. It is shown that, in a model of homogeneous turbulence in which each wavenumber of the energy spectrum predominantly receives contributions from eddies with roughly homogeneous amplitudes and coherence lengths, the p.d.f. would be slightly sub-Gaussian. This agrees with the available experimental evidence.

scholarly journals On the ubiquitous notion of mean in probability and statistics

2019 ◽  
Vol 2 (2) ◽  
pp. 35-41
Author(s):  
Aboubakar MAITOURNAM
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Law Of Large Numbers ◽  
Probability Distributions ◽  
Basic Notion ◽  
The Core ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Essential Parameter ◽  
Probability And Statistics

In probability and statistics, the basic notion of probability of an event can be expressed as a mathematical expectation. The latter is a theoretical mean and is an essential parameter of most probability distributions, in particular of the Gaussian distribution. Last but not least, the notion of mean is at the core of two main theorems of probabilities and statistics, that is : the law of large numbers and the central limit theorem. Whether it is a theoretical or empirical version, the concept of mean is omnipresent in probability and statistics, is consubstantial to these two disciplines and is a bridge between randomness and determinism.

scholarly journals Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 880
Author(s):  
Igoris Belovas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Rate Of Convergence ◽  
Limit Theorems ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Constructive Proof ◽  
The Central Limit Theorem ◽  
Local Limit Theorems

In this research, we continue studying limit theorems for combinatorial numbers satisfying a class of triangular arrays. Using the general results of Hwang and Bender, we obtain a constructive proof of the central limit theorem, specifying the rate of convergence to the limiting (normal) distribution, as well as a new proof of the local limit theorem for the numbers of the tribonacci triangle.

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