The Normal Gaussian Distribution

The constant C comes from the requirement that the sum of all the probabilities equals one. If N is large, we can approximate M as a continuous variable instead of something restricted to integral values (and this will be more realistic for chemical and physical applications). Then we have 

This last integral is identical to Equation 2.29, with the substitutions x = M and a = 

Now we can see how this distribution behaves for large values of N. For example, if we toss 10,000 coins, we have a = -JN = 100 in Equation 4.14. We would expect to get, on average, 5000 heads and 5000 tails (M = 0), and indeed the maximum of P(M) occurs at M = 0. But if M C 100, P(M) is only slightly smaller than P 0). Thus some deviations from exact equality are quite likely. 

We can calculate expectation values for this continuous distribution in much the same way as we calculated them in the last section for ten coin tosses. The generalization of Equation 4.8 for a continuous distribution is  

Equation 4.16 shows that the width of the distribution is proportional to the square root of the number of steps. So -JN (which is also the standard deviation a) provides a measure of the fluctuations of M from its average value of zero. After 10,000 random steps, we will be, on average, about VI 0,000 = 100 steps away from the starting point. But if we take 1,000,000 random steps (100 times more steps) we will only end up on average 1000 steps from the start (10 times as far) out of all of these steps, in effect 

Both the binomial and Poisson distributions apply to discrete variables, whereas most of the random variables involved in experiments are continuous. In addition, the use of discrete distributions necessitates the use of long or infinite series for the calculation of such parameters as the mean and the standard deviation (see Eqs. 2.47, 2.48, 2.52, 2.53). It would be desirable, therefore, to have a pdf that applies to continuous variables. Such a distribution is the normal or Gaussian distribution. 

Notice that this distribution, shown in Fig. 2.4, has a maximum at x = w, is symmetric around m, is defined uniquely by the two parameters a and m, and extends from x = —i to x = + . Equation 2.55 represents the shaded area under the curve of Fig. 2.4. In general, the probability of finding the value of x 

Three very important items associated with the Gaussian distribution are the following. 

Equation 2.62 indicates that 68.3 percent of the total area under the Gaussian is included between m — a and m + a. Another way of expressing this statement is to say that if a series of events follows the normal distribution, then it should be expected that 68.3 percent of the events will be located between m — a and m + O. As discussed later in Sec. 2.13, Eq. 2.62 is the basis for the definition of the standard error. 

The full width at half maximum (FWHM). The FWHM, usually denoted by the symbol F, is the width of the Gaussian distribution at the position of half of its maximum. The width F is slightly wider than 2 a (Fig. 2.4). The correct 

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The particle size distribution of ball-milled metals and minerals, and atomized metals, follows approximately the Gaussian or normal distribution, in most cases when the logarithn of die diameter is used rather dran the simple diameter. The normal Gaussian distribution equation is... 

The reason for calling equation 8.3-1 a "Gaussian diffusion model" is because it has the form of the normal/Gaussian distribution (equation 2.5-2). Concentration averages for long time intervals may be calculated by averaging the concentrations at grid elements over which the plume passes. 

The normal (Gaussian) distribution is the most frequently used probability function and is given by... 

A further consideration is that the value of the calculated nonlinearity will depend not only on the function that fits the data, we suspect that it will also depend on the distribution of the data along the X-axis. Therefore, for pedagogical purposes, here we will consider the situation for two common data distributions the uniform distribution and the Normal (Gaussian) distribution. 

The skewness k characterizes the asymmetry of the spectrum, whereas the excess k2 describes deviation of the levels or line density from the normal (Gaussian) distribution density. 

The normal Gaussian distribution df coarse HMX impact sensy is used to estimate the increased sensy of coarse HMX contg various amounts of fine ( <63 microns) airborne grit by a statistical modeled expt. The exptl results were then used in a BemoulUan confidence level eqtn to determine the sample size required to accurately estimate the sensy of any grit contg HMX sample with a K % level of confidence. A sample of the type of K table derived is shown for HMX in Table 5]... 

It should be emphasized that for the Markovian copolymers, the knowledge of these structure parameters will suffice for finding the probabilities of any sequences LZ, i.e., for a comprehensive description of the structure of the chains of such copolymers at their given average composition. As for the CD of the Markovian copolymers, for any fraction of Z-mers it is described at Z 1 by the normal Gaussian distribution with covariance matrix, which is controlled along with Z only by the values of structure parameters (Lowry, 1970). The calculation of their dependence on time and on the kinetic parameters of a reaction system enables a complete statistical description of the chemical structure of a Markovian copolymer. It is obvious therewith to which extent a mathematical modeling of the processes of the synthesis of linear copolymers becomes simpler when the sequence of units in their macromolecules is known to obey Markov statistics. 

For many cases, especially if the spreading is small, the spreading function can be approximated by the normal Gaussian distribution function G in the form ... 

In many instances, the normal (Gaussian) distribution best describes the observed pattern, giving a symmetrical, bell-shaped frequency distribution (p. 274) for example replicate measurements of a particular characteristic (e.g. rejjeated measurements of the end-point in a titration). 

At times, there is a need to compare the variances (or standard deviations) of two populations. For example, the normal t test requires that the standard deviations of the data sets being compared are equal. A simple statistical test, called the F test, can be used to test this assumption under the provision that the populations follow the normal (Gaussian) distribution. The F test is also used in comparing more than two means (see Section 7C) and in linear regression analysis (see Section 8C-2). 

The decision that there is nothing to report has a confidence limit of 1 — a, where a is a certain fraction of the normalized Gaussian distribution (Fig. 2.12). Take as an example a = 0.05. Then... 

The binomial distribution law correctly expresses this probability, but it is common practice to use either the Poisson distribution or the normal Gaussian distribution fimctions since both approximate the first but are much simpler to use. If the average number of counts is high (above 100) the Gaussian function may be used with no appreciable error. The probability for observing a measured value of total count N is... 

The initial assumption made in Eq. (53) concerning the normal distribution of micropore volume according to the parameter B is somewhat artificial. Perhaps, a more plausible assumption is to use the normal Gaussian distribution of the half width of slit-like micropores [114] ... 

The log-normal distribution gives a curve skewed towards the larger sizes, and it frequently gives a good representation of particle size distributions from precipitation and comminution processes. Furthermore, the log-normal distribution is often used because it overcomes the objection to the normal (Gaussian) distribution function which implies the existence of particles of negative size. 

Figure 1 The normal (Gaussian) distribution represented as (A) a frequency curve and (B) a cumulative frequency curve.

The Greek letter used to indicate the standard deviation of a population, defined as the square root of the variance, e.g., for the normal (Gaussian) distribution ... 

Y,i,k (Equation [8.26]) is the standard deviation of k values of Yj predicted by Equation [8.19a] for a known Xj and Vy i is the variance of the normal (Gaussian) distribution of replicate determinations assumed to underlie the small data sets usually obtained in analytical practice, as predicted from Equation [8.19a] for a chosen value of Xj. Clearly (sy j t) = Vy j = (sy j) in the limit k - 00, i.e. Vy i pjjji is the variance of the normal distribution assumed to describe the determinations of Yj. Later the quantity s(Yj) is used to denote a simple experimental determination (not prediction as for Sy j ) of the standard deviation of a set of replicate determinations of Yj for a fixed Xj in the calibration experiments (Equation [8.2] with Y replacing x). 

The statistical tests developed in the previous chapters have all assumed that the data being examined follow the normal (Gaussian) distribution. Some support for this assumption is provided by the central limit theorem, which shows that the sampling distribution of the mean may be approximately normal even if the parent population has quite a different distribution. However, the theorem is not really valid for the very small data sets (often only three or four readings) frequently used in analytical work. 

K 2> and constants. If specially required, selected values of p, q, r, and s can also be searched for by simultaneous regression estimation. The highest number of parameters should not exceed 16. Apart from this, a simulation can be performed while theoretical data calculated from assumed values of parameters (p, q, r, s), and K i are loaded with a random noise having the normal (Gaussian) distribution of errors. An arbitrary level of noise can be chosen by the user. 

The derivation of kinetic equations based on the Gaussian distribution of reactants among micelles is not as definitive as those of the Poisson distribution, because the variable of the Gaussian distribution is continuous, whereas the number of reactants in each micelle is an integer. When R is the value at which the distribution becomes maximal (i.e., the mean value of the Gaussian distribution), the normalized Gaussian distribution has the form... 

To simulate particle growth dynamics, the one-dimensional PBM with parameters, which are fisted in Table 10, have been used. The PSD of nuclei appearing due to the overspray and initial distribution of granules in apparatus have been distributed according to the normal (Gaussian) distribution. 

Thus the profile of a Dopper-broadened spectral line is given by the normalized Gaussian distribution (u -Up,A) shown in Fig.8.6. The half width (FWHM) is given by... 

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