Probability theory central limit theorem

R. A. Marcus Even though solvents and solvent-solute interactions or interactions with a protein can be very complicated and the resulting motion can be highly anharmonic, under a particular condition there can be a great simplification because of the many coordinates (perhaps analogous to the central-limit theorem in probability theory). 

Also using chemical space as a framework, Agrafiotis [118] presented a very fast method for diversity analysis on the basis of simple assumptions, statistical sampling of outcomes, and principles of probability theory. This method presumes that the optimal coverage of a chemical space is that of uniform coverage. The central limit theorem of probability theory... 

To obtain an expression for k p) we will assume that the process execution times for both algorithms (a) and (b) form a normal distribution, which is a reasonable assumption (according to the Central Limit Theorem from probability theory) when there is a large number of tasks per process. Assuming a normal distribution with mean /r and standard deviation a, the probability of a process execution time being below fj. + ka can be computed as 5 + jerf(k/V2), where erf denotes the error function. If there are p processes, the probability that all process execution times are below fi+ka is then given as [ -F jerf k/V2)]P. We need Eqs. 7.5 and 7.6 to be fairly accurate estimates for the maximum execution time, and we must therefore choose k such that... 

In general, we can say that the sum of a great number of random values is controlled by a Gaussian distribution of probabilities, like (6.16). This is one of the key ideas in probability theory. Due to its great importance, it was given a posh name, the central limit theorem (CLT). 

Disorder was introduced into this system by postulating a distribution of waiting times. A complementary extension of the theory may be made by considering a distribution of jump distances. It may be shown that, as a consequence of the central limit theorem, provided the single-step probability density function has a finite second moment, Gaussian diffusion is guaranteed. If this condition is not satisfied, however, then Eq. (105) must be replaced by... 

We have seen that a chain has a Gaussian property irrespective of the details of the model employed when the number n of the repeat units is large. This is a typical example of the central limit theorem in probability theory. 

The statistical average of the mean end-to-end distance is zero ((R) = 0), whereas its mean square deviation is linearly proportional to the number of monomers N R = Rf = b N). The radius of gyration is given as R = Rj/6. Equation [18] is the result of a central limit theorem from elementary probability theory. ... 

The Gaussian distribution is generally assumed to govern random experimental errors. The central limit theory of statistics gives some justification for this assumption. This theorem states that if a number of random variables (independent variables) xi,X2,. ..,x are all governed by probability distributions with finite means... 

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