The Simplest Discrete and Continuous Distributions

To illustrate the difference between values of sample estimates and population parameters, consider the ten groups of five numbers each as shown in the table. The sample means and sample standard deviations have been calculated from appropriate formulas and tabulated. Usually we could calculate no more than that these val- 

We can now see that our sample means in the ten groups scatter around the population mean. The mean of the ten group-means is 4.58, which is close to the population mean. The two would be identical if we had an infinite number of groups. Similarly, the sample variances scatter around the population variance, and their mean of 7.69 is dose to the population variance. 

What we have done in the table is to take ten random samples from the infinite population of numbers from 0 to 9. In this case, we know the population parameters so that we can get an idea of the accuracy of our sample estimates. 

From the table of random numbers take 20 different sample data with 10 random numbers. Determine the sample mean and sample variance for each sample. Calculate the average of obtained statistics and compare them to population parameters. 

In analyzing an engineering problem, we frequently set up a mathematical model that we believe will describe the system accurately. Such a model may be based on past experience, on intuition, or on a theory of the physical behavior of the system. 

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Furthermore, it should be borne in mind that the relationships obtained directly in the form of Eqs. (5.3.7) and (5.3.8) are used in the simplest manner only if the spectrum E(t ) is really continuous. The case, for which the distribution function of t contains discrete lines, must be considered specially ... 

One of the main motivations for the transition from LGA to LBM was the desire to replace the discrete collision rules with a continuous function known as the collision operator. The simplest and by now most admired form of the collision operator is the so-called quasilinear or single-relaxation time Bhatnagar-Gross-Kfook (BGK) collision operator. With this operator, the single-partide distribution function evolves to the equilibrium state via... 

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