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.
Highlights
Methodology to Study Collective Response TimesM. Isidro 1 , Esperanza Navarro-Pardo 2, * , Luisberis Velázquez-Abad 3 and
Introduction nal affiliationsThe Chi distribution is a widely known continuous probability distribution in Statistics.It represents the distribution of a random variable which is calculated as the Euclidean norm over k variables each following a standard Normal distribution [1], where k specifies the degrees of freedom
Monte Carlo simulations and non-linear fittings have been carried out to study the. Chi distribution when it is approximated by the square root of the sum of squared variables following non-standard Normal distributions with different values in the variance
Summary
M. Isidro 1 , Esperanza Navarro-Pardo 2, * , Luisberis Velázquez-Abad 3 and. Distribution with Unequal Variances in the Generating Gaussians
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