A Study of Understand the Weighted Maxwell- Boltzmann Distribution

To characterize the speed of a particle in an idealized gas, the Maxwell Distribution (MD) is frequently used in physics, particularly statistical mechanics. In addition, this distribution may be used to explain the distribution of values. of where and are the measurement errors in position coordinates of a particle in a 3-dimensional space. In 1859, Scottish physicist James Clerk Maxwell proposed this distribution. Ludwig Boltzmann, a German physicist, expanded Maxwell's finding in 1871 to describe the distribution of energy among molecules. One way to think about the speed of a particle travelling across three-dimensional space is to think of it as a set of independent and normally distributed random variables, each having a mean and variance equal to one-third of the rate parameter. However, if a particle travels in a two-dimensional space rather than a three-dimensional one, the Rayleigh distribution better describes its speed. A particle's kinetic energy (E), which is inversely proportional to its velocity, may be determined using MD.by the formula , , provided the distribution of speed is known. The p.d.f. of MD is given by where denotes the mass of particle, the Boltzmann‘s constant and thermodynamic temperature. Re-parameterizing equation (2.1) by , we have the resulting p.d.f. as given in (2.2).

MD has several uses in both physics and chemistry. Pressure and diffusion are only two of the many basic phenomena of gases that may be explained using MD. The distribution of velocities, energies, and magnitudes of molecular momenta is commonly referred to as this. When it came to researching randomness in physical and chemical sciences, MD was key. An investigation of its statistical qualities may be done in a variety of different ways in Statistics

Consider the weight functions, where the weight parameter is. Therefore, Now, using the definition of weighted distribution given by (1.53), we get the p.d.f. of weighted

We use as the notation for denoting a random variable following WMD with rate and weight parameter in the rest of chapter. C.d.f., reliability function and hazard rate of WMD are respectively given by (2.5), (2.6) and (2.7).

Weighted distribution may be traced back to the work of Fisher (1934), in which it is examined how ascertainment procedures might affect how recorded observations are distributed. Rao (1965) used it to represent statistical data when conventional distributions were determined to be inappropriate for the task. Rao then refined and formalized it in generic terms. Consequently, Fisher (1934) and Rao (1965) are credited with first pondering non-experimental, non-replicated, and non-random observation conditions before introducing the notion of weighted distribution. Observations are only recorded when they are encountered, and this may be shown best via encounter sampling. Conditions where observations are collected with probability proportional to some weight function w may be described more broadly (x). Every observation must be given an equal opportunity to be recorded if we are to get the true distribution of the seen data while researching real world random occurrences. If this does not occur, the recorded data will not have the original distribution. Rao (1965) came up with the idea of weighted distributions as a unifying technique for the issue of model design and statistical inference after researching comparable cases.

Let be an ordered sample of size from WTPFD (). As a result, the logarithmic likelihood function is equal to

From Figure 3.5, it is clear that the gradient given by (3.29) is negative for all possible values of , which implies that (3.28) is a decreasing function w.r.t. . Therefore, M.L.E. of under the restriction nis given by , whereas, M.L.E.‘s for the rest of three parameters and are obtained by solving simultaneously the equation (3.30), (3.31) and (3.32). the following system of equations is obtained by equating the gradients of (3.28) w.r.t. and to zero. The above system of equations is non-linear. Therefore, to obtain the estimates in closed form is very tedious. Thus, R programming is used to find the required estimates of parameters after the fitting of WTPFD to some considered data sets.

Here, the structural characteristics of WMD have been described. m.g.f. and characteristic function expressions for rth instant about origin and mean are provided. Also included are expressions for rth random variable is given by

Proof.By the definition of moment about origin we have Theorem 2.2.Square of sample coefficient of variation is asymptotically unbiased estimator of the square of population coefficient of variation, i.e., where and are respectively the mean and variance of a sample. Proof.Let be a random sample of size with and variance drawn from . Therefore, Using (2.19) and (2.20), we obtain Applying both sides, we get Theorem 2.3. The m.g.f. and characteristic function of are respectively given by (2.21) and (2.22).

Proof. From the definition of m.g.f. we have Also, we know that . Therefore

Maximal likelihood estimation and the technique of moments are used to estimate WMD parameters in

Let be a random sample of size from . Therefore, it‘s likelihood functions is given by (2.31). Log likelihood function is given by Differentiating log likelihood function partially with respect to and we get the following two gradients: On equating the derived gradients to zero and reducing them to simplified forms, we obtain following system of two equations: Substituting (2.35) in (2.36), we get It is impossible to obtain the M.L.E. of w by solving (2.37) manually for . Therefore, an estimate of is computed numerically by using the following code written in Wolfram Mathematics programming language after supplying a guess value say and a data set say . After obtaining the numerical estimate of say , it is substituted in (2.35) in order to have the corresponding estimate of which is given by Now, on using (1.45) given in Section 1.2.10.2.1, we get Fisher information matrix associated with and is given by Where is known as digamma or Psi function. Thus, the asymptotic 100 (1-)% confidence interval for is given by Since, therefore we can write Therefore, 95% confidence interval for and is respectively given by where and represents the element belonging to row and column of inverse of Fisher information matrix.

A wide range of WMD-related characteristics have been investigated and explored in depth. The validity of WMD in statistical modelling is shown using three real-world data sets, including intensity, noise, wear, and a simulated one. By using inverse sampling, a simulated dataset is created. After fitting WMD to the input data, parameter moment estimates, maximum likelihood estimates, Fisher information matrices, and their inverses are calculated. When considering WMD, several statistical metrics such as the accuracies of weapons of mass destruction have been calculated. We already know that the distribution with the lowest AIC, BIC, and AICc is deemed to be the best fit. WMD is shown to have the lowest AIC, BIC, and AICc, followed by ABMD, LBMD, circumstances..

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Research Scholar, Sunrise University, Alwar, Rajasthan