The Gaussian Error Function

The centerpiece of statistical treatment of data is the Gaussian Error function which gives mathematical expression for the behavior of random 

Note that this graph approximates the Gaussian curve, which is a continuous function, so that it is comprised of an infinite number of points. Repeat the frequency distribution using 0.05 as bin interval. With the smaller number of bins, the distribution will conform better to the shape of the error curve. Why  

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The second term in the braces is the Gaussian error function, x = cA/cA, and fluorescence quantum yield of A when it is either excited directly or when energy transfer from D to A following excitation of D is complete at high concentration of A. This function and its counterpart, c/>D(cA)/0D° = 1 a aV a representing the concentration quenching of D by A, are plotted in Figure 2.16. 

The sign of equality holds only for one definite distribution (proper function), viz. the Gaussian error function, which occurs in the case of the linear oscillator (see A])pen(lix XII, p. 282 Appendix XVI, p. 295 and Appendix XXXII, p. 343). 

The functions erf (x) and erfi(x) = erf(ix) fi denote the Gaussian error function and the imaginary Gaussian error function, respectively. The remaining integral can be solved by series expansion so that the final expression for the mean first passage time takes the form... 

The Gaussian error function correctly describes the variations to be expected in a population of values of infinite size, an obviously unattainable level of replication in practice. With the general guidelines embodied in this function, a body of statistical rules applicable to small sets of data has been developed. 

The second term is the Gaussian error function, erf (y), and may be evaluated from mathematical tables. Equation (37) may be expanded as a series ... 

This distribution has the form of the Gaussian error function and is spherically symmetrical about the origin, where the value is a maximum (Figure 3.4). The most probable end-to-end length is not, however, zero, as the probability that Q falls within an elemental volume situated between r and (r + dr) from the origin, irrespective of direction, is the product of the probability distribution p r) and the volume of the concentric shell, dr. The overall probability is then... 

The function given by Equation 70 is the Gaussian error function. At time t, the initially thin slab of charge carriers has attained a bell-shape distribution (see Figure 19a). TTie broadening is described by... 

For a constant smface composition boundary condition, the solution to Pick s second law (Equation 5.4b) is Equation 5.5, which involves the Gaussian error function (erf). 

Trial wavefunctions are usually constructed by linear combination of Gaussian error functions that are convenient to integrate. The results can be of predictive value and such calculations have become everyday tools for chemists in all branches of chemistry, to guide experiments and not least to rule out untenable hypotheses. This is a remarkable achievement that seemed to be out of reach a few decades ago. Still, simple qualitative models that are amenable to perturbation theory are required to understand and predict trends in a series of related compounds. Our goal here is to describe the minimal quantum mechanical models that can still provide a useful qualitative description of electronically excited states, their electronic stmcture and their reactivity. Such models also provide a language to convey the results of state-of-the-art, but essentially black-box ab initio calculations. 

We can write down one solution of the equation at once, namely, the so-called Gaussian error function... 

This is also a Gaussian error function, with its maximum at x — Xo and with uncertainty Ax given by Equation 53-5. It is seen that the wave function x corresponds to the value h/2ir for the product of the uncertainties Ax and Apx at the time t = 0, this value being of the order of magnitude h, as stated at the beginning of the section. 

The name error function is chosen because of its frequent use in probability calculations involving the Gaussian probability distribution. Another form giving the same information is the normal probability integral ... 

The Quantities a and U. Samples prepared by an unperturbed anionic polymerization process have a distribution function, which, as derived by Bohm (I), corresponds to a Gaussian error function. The same in general is the case for fractions produced by precipitation and/or solution. The distribution function then is characterized by two parameters, the variance a and the number average degree of polymerization Pn. On the other hand, a given distribution can be characterized by the nonuniformity (Uneinheitlichkeit in German) C7, defined by the equation... 

The error function can be viewed as a partial area under the Gaussian distribution curve (2/n )exp -u ). It has the value 0 at x = 0, and a value of 1 at X = oo. The latter case corresponds to the full area under the distribution curve. A related expression is the complementary error function erfc x, which is defined as... 

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