Monte Carlo Simulation of a Modified Chi Distribution with Unequal Variances in the Generating Gaussians. A Discrete Methodology to Study Collective Response Times
| Autor: | Pedro Fernández-de-Córdoba, Esperanza Navarro-Pardo, Juan Carlos Castro-Palacio, Luisberis Velázquez-Abad, José M. Isidro |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Distribution (number theory)
Chi distribution Keywords: Chi distribution General Mathematics Monte Carlo method Degrees of freedom (statistics) 050109 social psychology 02 engineering and technology Maxwell-Boltzmann distribution Normal distribution symbols.namesake 0202 electrical engineering electronic engineering information engineering Computer Science (miscellaneous) 0501 psychology and cognitive sciences discrete model Statistical physics Engineering (miscellaneous) lcsh:Mathematics 05 social sciences Variance (accounting) lcsh:QA1-939 Maxwell–Boltzmann distribution Psicologia symbols reaction times 020201 artificial intelligence & image processing Random variable |
| Zdroj: | Castro Palacio, Juan Carlos Isidro San Juan, José María Navarro Pardo, Esperanza Velázquez Abad, Luisberis Fernández de Córdoba, Pedro 2021 Monte Carlo Simulation of a Modified Chi Distribution with Unequal Variances in the Generating Gaussians. A Discrete Methodology to Study Collective Response Times Mathematics 9 77 RODERIC. Repositorio Institucional de la Universitat de Valéncia instname RODERIC: Repositorio Institucional de la Universitat de Valéncia Mathematics Volume 9 Issue 1 Mathematics, Vol 9, Iss 77, p 77 (2021) |
| Popis: | The Chi distribution is a continuous probability distribution of a random variable obtained from the positive square root of the sum of k squared variables, each coming from a standard Normal distribution (mean = 0 and variance = 1). The variable k indicates the degrees of freedom. The usual expression for the Chi distribution can be generalised to include a parameter which is the variance (which can take any value) of the generating Gaussians. For instance, for k = 3, we have the case of the Maxwell-Boltzmann (MB) distribution of the particle velocities in the Ideal Gas model of Physics. In this work, we analyse the case of unequal variances in the generating Gaussians whose distribution we will still represent approximately in terms of a Chi distribution. We perform a Monte Carlo simulation to generate a random variable which is obtained from the positive square root of the sum of k squared variables, but this time coming from non-standard Normal distributions, where the variances can take any positive value. Then, we determine the boundaries of what to expect when we start from a set of unequal variances in the generating Gaussians. In the second part of the article, we present a discrete model to calculate the parameter of the Chi distribution in an approximate way for this case (unequal variances). We also comment on the application of this simple discrete model to calculate the parameter of the MB distribution (Chi of k = 3) when it is used to represent the reaction times to visual stimuli of a collective of individuals in the framework of a Physics inspired model we have published in a previous work. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|