Random central limit theorem

The random manner by whieh the inherent inaeeuraeies within the proeess are generated produees a pattern of variation for the dimension that resembles the Normal distribution, as diseussed in Chapter 2. As a first supposition then in the optimization of a toleranee staek with number of eomponents, it is assumed that eaeh eomponent follows a Normal distribution, therefore giving an assembly toleranee with a Normal distribution. It is also a good approximation that if the number of eomponents in the staek is greater than 5, then the final assembly eharae-teristie will form a Normal distribution regardless of the individual eomponent distributions due to the central limit theorem (Misehke, 1980). 

The Central Limit Theorem gives an a priori reason for why things tend to be normally distributed. It says the sum of a large number of independent random distributions having finite means and variances is normally distributed. Furthermore, the mean of the resulting distribution the sum of the individual means the combined variance is the sum of the individual variance.. ... 

Our next result concerns the central limit theorem, which places in evidence the remarkable behavior of the distribution function of when n is a large number. We shall now state and sketch the proof of a version of the central limit theorem that is pertinent to sums of identically distributed [p0i(x) = p01(a ), i — 1,2, ], statistically independent random variables. To simplify the statement of the theorem, we shall introduce the normalized sum s defined by... 

The Central Limit Theorem.—If 4>i,4>a, we identically distributed, statistically independent random variables having finite mean and variance, then... 

The central limit theorem thus states the remarkable fact that the distribution function of the normalized sum of identically distributed, statistically independent random variables approaches the gaussian distribution function as the number of summands approaches infinity—... 

Essentially the same argument used above enables one to prove an important multidimensional version of the central limit theorem that applies to sums of independent random vectors. A -dimensional random vector is simply a group of k random variables,... 

If U0 and U1 were the functions of a sufficient number of identically distributed random variables, then AU would be Gaussian distributed, which is a consequence of the central limit theorem. In practice, the probability distribution Pq (AU) deviates somewhat from the ideal Gaussian case, but still has a Gaussian-like shape. The integrand in (2.12), which is obtained by multiplying this probability distribution by the Boltzmann factor exp (-[3AU), is shifted to the left, as shown in Fig. 2.1. This indicates that the value of the integral in (2.12) depends on the low-energy tail of the distribution - see Fig. 2.1. 

According to the central limit theorem, if one sums up random variables which are drawn from any (but the same for all variables) distribution (as long as this distribution has finite variance), then the sum is distributed according to a Gaussian. In this... 

The physical and conceptual importance of the normal distribution rests on one unique property the sum of n random variables distributed with almost any arbitrary distribution tends to be distributed as a normal variable when n- oo (the Central Limit Theorem). Most processes that result from the addition of numerous elementary processes therefore can be adequately parameterized with normal random variables. On any sort of axis that extends from — oo to + oo, or when density on the negative side is negligible, most physical or chemical random variables can be represented to a good approximation by a normal density function. The normal distribution can be viewed a position distribution. 

When we are dealing with samples rather than populations, we cannot use the standard normal deviate, Z, to make predictions since this requires knowledge of the population mean and variance or standard deviation. In general, we do not know the value of these parameters. However, provided the sample is a random one, its mean 5 is a reliable estimate of the population mean p, and we can use the central limit theorem to provide an estimate of o. This esti mate, known as the standard error of the mean, is given by ... 

Exercise. Apply the central limit theorem to the random walk in 4, comp. (4.7). Compare the result with an explicit calculation as above. 

There is no a priori reason to doubt that the Central Limit Theorem, and consequently the normal distribution concept, applies to trace element distribution, including Sb and Ba on hands in a human population, because these concentrations are affected by such random variables as location, diet, metabolism, and so on. However, since enough data were at hand (some 120 samples per element), it was of interest to test the normal distribution experimentally by examination of the t-Distribution. The probability density plots of 0.2 and 3 ng increments for Sb and Ba, respectively, had similar appearances. The actual distribution test was carried out for Sb only because of better data due to the more convenient half life of 122Sb. After normalization, a "one tail" test was carried out. 

Frequentist methods are fundamentally predicated upon statistical inference based on the Central Limit Theorem. For example, suppose that one wishes to estimate the mean emission factor for a specific pollutant emitted from a specific source category under specific conditions. Because of the cost of collecting measurements, it is not practical to measure each and every such emission source, which would result in a census of the actual population distribution of emissions. With limited resources, one instead would prefer to randomly select a representative sample of such sources. Suppose 10 sources were selected. The mean emission rate is calculated based upon these 10 sources, and a probability distribution model could be fit to the random sample of data. If this process is repeated many times, with a different set of 10 random samples each time, the results will vary. The variation in results for estimates of a given statistic, such as the mean, based upon random sampling is quantified using a sampling distribution. From sampling distributions, confidence intervals are obtained. Thus, the commonly used 95% confidence interval for the mean is a frequentist inference... 

As we saw in Section 3.1.1, the familiar bell-shaped curve describes the sampling distributions of many experiments. Many distributions encountered in chemistry are approximately normal [3], Regardless of the form of the parent population, the central limit theorem tells us that sums and means of samples of random measurements drawn from a population tend to possess approximately bell-shaped distributions in repeated sampling. The functional form of the curve is described by Equation 3.19. 

The random nature of the second term on the right assures us, through the central limit theorem, that it contributes an effective diffusion term to zone spreading [11]. Thus, this term must have the equivalent form... 

The goal in an NMR experiment is to maximize S/N, and this can be achieved in several ways. One method is to use higher rf power and a series of rf filters to remove some of the noise. While this technique results in some improvement, care must be taken to avoid saturation problems (see Section 2.3). A far better improvement can be obtained by relying on a result of the central limit theorem from information theory. This theorem tells us that if the spectrum is scanned n times and the resulting data are added together, signal intensity will increase directly with n while noise (being random) will only increase by Vn ... 

Equations (4.3-4) and (4.3-5) are the first of several important limit theorems that establish conditions for asymptotic convergence to normal distributions as the sample space grows large. Such results are known as central limit theorems, because the convergence is strongest when the random variable is near its central (expectation) value. The following two theorems of Lindeberg (1922) illustrate why normal distributions are so widely useful. 

These central limit theorems are also relevant to physical models based on random processes. These theorems tell us that normal distributions can arise in many ways. Therefore, the occurrence of such a distribution tells very little about the mechanism of a process it indicates only that the number of random events involved is large. 

The Central-Limit Theorem states that the sampling distribution of the mean, for any set of independent and identically distributed random variables, will tend toward the normal distribution, equation (3.17), as the sample size becomes large. ... 

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