Author
Listed:
- Juan Carlos Castro-Palacio
(Grupo de Modelización Interdisciplinar, Instituto Universitario de Matemática Pura y Aplicada, InterTech, Universitat Politècnica de València, E-46022 Valencia, Spain
Current affiliation: Department of Electrical Engineering, Electronics, Automation, and Applied Physics, Technical University of Madrid. Ronda de Valencia, 3, 28012 Madrid, Spain.)
- J. M. Isidro
(Grupo de Modelización Interdisciplinar, Instituto Universitario de Matemática Pura y Aplicada, InterTech, Universitat Politècnica de València, E-46022 Valencia, Spain)
- Esperanza Navarro-Pardo
(Grupo de Modelización Interdisciplinar, Departamento de Psicología Evolutiva y de la Educación, InterTech, Universitat de València, E-46010 Valencia, Spain)
- Luisberis Velázquez-Abad
(Departamento de Física, Universidad Católica del Norte, Antofagasta 0610, Chile)
- Pedro Fernández-de-Córdoba
(Grupo de Modelización Interdisciplinar, Instituto Universitario de Matemática Pura y Aplicada, InterTech, Universitat Politècnica de València, E-46022 Valencia, Spain)
Abstract
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.
Suggested Citation
Juan Carlos Castro-Palacio & J. M. Isidro & Esperanza Navarro-Pardo & Luisberis Velázquez-Abad & Pedro Fernández-de-Córdoba, 2020.
"Monte Carlo Simulation of a Modified Chi Distribution with Unequal Variances in the Generating Gaussians. A Discrete Methodology to Study Collective Response Times,"
Mathematics, MDPI, vol. 9(1), pages 1-10, December.
Handle:
RePEc:gam:jmathe:v:9:y:2020:i:1:p:77-:d:473014
Download full text from publisher
Corrections
All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:77-:d:473014. See general information about how to correct material in RePEc.
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
We have no bibliographic references for this item. You can help adding them by using this form .
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .
Please note that corrections may take a couple of weeks to filter through
the various RePEc services.