Central limit theorem applications

Various sets of mathematical conditions for the rigorous validity of the central limit theorem can be found in the textbooks. 0 The physicist is better served with a qualitative insight into the scope of its applicability. For this purpose we add a few comments. 

The usual assumptions leading to the normal error probability function are those required for the validity of the central limit theorem The assumptions leading to this theorem are sufficient but not always altogether necessary the normal error probability function may arise at least in part from circumstances different from those associated with the theorem. The factors that in fact determine the distribution are seldom known in detail. Thus it is common practice to assume that the normal error probability function is applicable even in the absence of valid a priori reasons. For example, the normal error probability function appears to describe the 376 measurements of Fig. 3 quite well. However, a much larger number of measurements might make it apparent that the true probability function is slightly skewed or flat topped or double peaked (bimodal), etc. 

Actually, least squares is often applied in cases where it is not known with any certainty that measurements of jy conform to a normal distribution or even when it is in fact known that they do not conform to a normal distribution. Does this destroy the applicability of the maximum-likelihood criterion The answer is, not necessarily. The central-limit theorem is discussed briefly in Chapter 11. Simply stated, it says that the sum (or average) of a large number of measurements conforms very nearly to a normal distribution, regardless of the distributions of the individual measurements, provided that no one measurement contributes more than a small fraction to the sum (or average) and that the variations in the widths of the individual distributions are within reasonable bounds. (As we shall see, the average of a group of numbers is a special case of a least-squares determination.)... 

Bhatacharya, R., and V.K. Gupta. 1990. Application of the central limit theorem to solute transport in saturated porous media from kinetic to field scales, p. 97-124. In J.H. Cushman (ed.) Dynamics of fluids in hierarchical porous media. Acad. Press, New York. 

This result is known as the central limit theorem and serves to emphasize the importance and applicability of the normal distribution function in statistical data analysis since non-normal data can be normalized and can be subject to basic statistical analysis. ... 

Consider a stream of risky cash flows Aj occurring at the ends of periods 1, 2,. .. j,. .. /V. The project life N and the discount rate i are known with certainty. The only stochastic variable here is the amount of the cash flow. The resulting PW is a random variable with mean given by Eq. (8) and, assuming independent cash flows, with variance given by Eq. (10). Under some general conditions application of the central limit theorem leads to the result that... 

The terms Aj may have essentially any distribution. As a general rule of thumb, if the A s are approximately normally distributed, then the central limit theorem is a very good approximation when > 4. If the distribution of the A s has no prominent mode(s), that is, approximately uniformly distributed, then N > 12 is a reasonable rule of thumb for applicability of the central limit theorem. 

Equation (1.20) has many applications in this book, such as the diffusion motion of a Brownian particle (Appendix 3.D) and the probability distribution of the end-to-end vector of a long polymer chain. The latter case will be studied in this chapter. Essentially, (x ) / jjj gq (1.20) can be regarded as the mean projection of an independent segment (bond) vector in one of the three coordinate directions (i.e. x, y or z all three directions are equivalent). As long as the considered polymer chain is very long, we can always apply the central limit theorem, regardless of the local chemical structure. 

Thus, as given by Eq. (1.42), the probability distribution function for the end-to-end vector R is Gaussian. The distribution has the unrealistic feature that R can be greater than the maximum extended length Nb of the chain. Although Eq. (1.42) is derived on the freely jointed chain model, it is actually valid for a long chain, where the central limit theorem is applicable, except for the highly extended states. 

Normality assumption Most traditional SPC tools are based on the assumption that the process output characteristic is normally distributed, among which Shewhart control charts and multivariate control charts. In some cases, the central limit theorem can be used to justify approximate normality when monitoring means, but in numerous cases normality is an untenable assumption, and one is unwilling to use another parametric model. A number of nonparametric methods are available in these cases. As data availability increases, nonparametric methods seem especially useful in multivariate applications where most methods proposed thus far rely on normality. 

This is equivalent to the statement that the sum over a large number of subsystems gives the average value, something that would be expected in case the central limit theorem (CLT) is applicable. This generally is the case if quantities like M for the sub-blocks are independent and uncorrelated random variables. 

In this section, the algebra of random variables will be rehearsed from the viewpoint of error propagation. Convolution, central limit theorem, as well as random sums (i.e., sums of a random number of random variables) are also included here because of their importance in nuclear applications. 

Industrial plant measurements are not normally distributed. However, for large subgroup sizes n > 25), X is approximately normally distributed even if X is not, according to the famous Central Limit Theorem of statistics (Montgomery and Runger, 2007). Fortunately, modest deviations from normality can be tolerated. In addition, the standard SPC techniques can be modified so that they are applicable to certain classes of nonnormal data (Jacobs, 1990). 

There are many different ways to treat mathematically uncertainly, but the most common approach used is the probability analysis. It consists in assuming that each uncertain parameter is treated as a random variable characterised by standard probability distribution. This means that structural problems must be solved by knowing the multi-dimensional Joint Probability Density Function of all involved parameters. Nevertheless, this approach may offer serious analytical and numerical difficulties. It must also be noticed that it presents some conceptual limitations the complete uncertainty parameters stochastic characterization presents a fundamental limitation related to the difficulty/impossibility of a complete statistical analysis. The approach cannot be considered economical or practical in many real situations, characterized by the absence of sufficient statistical data. In such cases, a commonly used simplification is assuming that all variables have independent normal or lognormal probability distributions, as an application of the limit central theorem which anyway does not overcome the previous problem. On the other hand the approach is quite usual in real situations where it is only possible to estimate the mean and variance of each uncertainty parameter it being not possible to have more information about their real probabilistic distribution. The case is treated assuming that all uncertainty parameters, collected in the vector d, are characterised by a nominal mean value iJ-dj and a correlation =. In this specific... 

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