A Central Limit Theorem

Let us consider a stochastic variable, Yx, which is the deviation from the average of the arithmetic mean of N statistically independent measurements of a stochastic variable, X. Yn may be written as 

We want to obtain the probability distribution function of Yn, PY,N y)-First, the characteristic function, fz k, N), for the stochastic variable, Z w, can be written as 

Regardless of the form of Px x), if it has finite moments, the average of a large number of statistically independent measurements of X will be a Gaussian function centered at (x), with a standard deviation which is smaller than the standard deviation of the probability distribution of a single measurement X, ax, by a factor of N°-.  

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. 

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The connection between the stochastic cell model and the usual deterministic model of reaction-diffusion systems were given by Kurtz for the homogeneous case in the same spirit. To give the relationship among a Markovian jump process and the solution p(r, i) of the deterministic model a law of large numbers and a central limit theorem hold (Arnold Theodoso-pulu, 1980 Arnold 1980). 

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. ... 

Sample Distributions and the Central Limit Theorem Let s return to the problem of determining a penny s mass to explore the relationship between a population s distribution and the distribution of samples drawn from that population. The data shown in Tables 4.1 and 4.10 are insufficient for our purpose because they are not large enough to give a useful picture of their respective probability distributions. A better picture of the probability distribution requires a larger sample, such as that shown in Table 4.12, for which X is 3.095 and is 0.0012. 

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.. ... 

The Characteristic Function.—The calculation of moments is often quite tedious because of difficulties that may be encountered in evaluating the pertinent integrals or sums. This problem can be simplified quite often by calculation of the so-called characteristic function of the distribution from which, as we shall see, all moments can be derived by means of differentiation. This relationship between the characteristic function and moments is sufficient reason for studying it at this time however, the real significance of the characteristic function will not become apparent until we discuss the central limit theorem in a later section. 

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... 

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,... 

Notice that those distribution functions that satisfy Eq. (4-179) still constitute a convex set, so that optimization of the E,R curve is still straightforward by numerical methods. It is to be observed that the choice of an F(x) satisfying a constraint such as Eq. (4-179) defines an ensemble of codes the individual codes in the ensemble will not necessarily satisfy the constraint. This is unimportant practically since each digit of each code word is chosen independently over the ensemble thus it is most unlikely that the average power of a code will differ drastically from the average power of the ensemble. It is possible to combine the central limit theorem and the techniques used in the last two paragraphs of Section 4.7 to show that a code exists for which each code word satisfies... 

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... 

So how does this help us determine n As we know from our previous discussion of the Central Limit Theorem [2], the standard deviation of a sample from a population decreases from the population standard deviation as n increases. Thus, we can fix fi0 and yua and adjust the a and [3 probabilities by adjusting n and the critical value. 

In Sect. 7.4.6, we discussed various stochastic simulation techniques that include the kinetics of recombination and free-ion yield in multiple ion-pair spurs. No further details will be presented here, but the results will be compared with available experiments. In so doing, we should remember that in the more comprehensive Monte Carlo simulations of Bartczak and Hummel (1986,1987, 1993,1997) Hummel and Bartczak, (1988) the recombination reaction is taken to be fully diffusion-controlled and that the diffusive free path distribution is frequently assumed to be rectangular, consistent with the diffusion coefficient, instead of a more realistic distribution. While the latter assumption can be justified on the basis of the central limit theorem, which guarantees a gaussian distribution for a large number of scatterings, the first assumption is only valid for low-mobility liquids. 

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. 

The SEM is quite a bit smaller than the SD, making it very attractive to use in reporting data. This size difference is because the SEM actually is an estimate of the error (or variability) involved in measuring the means of samples, and not an estimate of the error (or variability) involved in measuring the data from which means are calculated. This is implied by the Central Limit Theorem, which tells us three major things. 

If two Gaussian functions are convolved, the result is a gaussian with variance equal to the sum of the variances of the components. Even when two functions are not Gaussian, their convolution product will have variance equal to the sum of the variances of the component functions. Furthermore, the second moment of the convolution product is given by the sum of the second moments of the components. The horizontal displacement of the centroid is given by the sum of the component centroid displacements. Kendall and Stuart (1963) and Martin (1971) provide helpful additional discussions of the central-limit theorem and attendant considerations. 

The central-limit theorem (Section III.B) suggests that when a measurement is subject to many simultaneous error processes, the composite error is often additive and Gaussian distributed with zero mean. In this case, the least-squares criterion is an appropriate measure of goodness of fit. The least-squares criterion is even appropriate in many cases where the error is not Gaussian distributed (Kendall and Stuart, 1961). We may thus construct an objective function that can be minimized to obtain a best estimate. Suppose that our data i(x) represent the measurements of a spectral segment containing spectral-line components that are specified by the N parameters... 

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 ... 

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. 

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). 

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