Monte Carlo Simulation of a Modified Chi Distribution with Unequal Variances in the Generating Gaussians. A Discrete Methodology to Study Collective Response Times

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.

Human Reaction Times: Linking Individual and Collective Behaviour Through Physics Modeling

“…(3) i leads to a Maxwell-Boltzmann distribution of | v i |, even if in our case, the variances of the three components are not all equal [40]. As already mentioned, we have obtained a coefficient of determination of R 2 = 0.88 using 24, 192 experimental reaction times.…”

“…The parameters k B , m and T are respectively the Boltzmann constant, particle mass and temperature of the system in thermodynamic equilibrium. The MB distribution is a Chi distribution with three degrees of freedom and equal variances in general [40]. The entropy of a thermodynamic system is maximal in equilibrium.…”

Evaluation and Mathematical Analysis of a Four-Dimensional Lotka–Volterra-like Equation Designed to Describe the Batch Nisin Production System

“…,18. In what follows, we will establish, by means of a Monte Carlo study [28,29], which conditions must be satisfied by the parameters defining the matrix A and the vector b so that the Lotka-Volterra equation has a single asymptotically stable equilibrium point with all its positive (and non-zero) coordinates.…”

“…For the culture pH (Q(t) = pH(t)) and total nitrogen (Q(t) = TN(t)) decrease, we modified the logistic decline equation presented by Goudar et al [28] as follows: For biomass (X(t)) production, the logistic equation presented by Goudar et al [30] was used by considering that the death cell rate was zero, as follows:…”