Normal probability integral

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 ... 

This function is the integrally normalized probability for each water molecule being oriented such that it makes an angle B between its OH bond vectors and the vector from the water oxygen to the carbon atom. This function is calculated for those molecules within 4.9 A of the carbon atom (nearest neighbors), as this distance marks the first minimum in the pair distribution function for that atom. The curve in Figure 10 is typical for hydrophobic hydration (22). 

The normal probability function as expressed by Eq. (14) is useful in theoretical treatments of random errors. For example, the normal probability distribution function is used to establish the probability Hthat an error is less than a certain magnitude 8, or conversely to establish the limiting width of the range, —8 to 8, within which the integrated probability P, given by... 

The Monte Carlo integration kernel fi(R) is the normalized probability distribution used to sample conformation space. For example, Zc [Eq. (1.2)] is numerically integrated as... 

In this case the reproducibility model of Aq is simply a normal probability distribution function p(Aq) under the null hypothesis, with mean = 0 and standard deviation = 7. The parameter for testing Hq is the P-value, or, as we call it here, the similarity index (SI), for our simple library search system defined as the integral of the reproducibility function, in this case a symmetrical Gaussian curve ... 

P(r) is the probability per unit distance that a small particle will be found at distance r from the center, that is. within a spherical shell of volume 4nr dr. Hence, P r) = 4nr dr. If P(r) were normalized, the integral in the numerator would represent the average value of r. so A/ times that integral replaces the sum. The denominator enforces normalization. Hence... 

In the continuous case, we suppose the sample space is the Euclidean space R ", and assume there is a (normalized) probability measure d/z defined by the density p which is a non-negative function. In this case the standard event space F is then typically taken to be the Borel a-algebra of subsets of R" which includes open balls and countable unions, countable intersections or relative complements of open balls in R . The measure of the set can be defined by Lebesgue integration... 

Now let s compute P. (If we had not made Stirling s approximation, our derivation above would have led to P, but mired in mathematical detail. The following is an easier way to get it.) To have a properly normalized probability distribution, you need the integral o er all the probabilities to equal one. To find P, integrate ... 

The function p(AG ) is the density of states for which the reaction coordinate is the same for DA to D A and the energy change in the reaction AE) is zero. Its pre-exponential factor, AnAksT) ", normalizes the integrated probability of finding a given value of AE, which is a Gaussian function of AE (Eq. 4.56). The density of states has dimensions of reciprocal energy. 

The doubly truncated normal probability density function (pdf) is normally distributed on the interval from a to b and takes on a value of 0 elsewhere, (Lee 1979). Since a valid pdf must integrate to 1 over its range, the doubly truncated normal pdf must be normalized. Figure 7, and is given by formula... 

Breitung K (1984) Asymptotic approximations for multinormal integrals. J Eng Mech 110(3) 357-366 Chen X, Lind NC (1983) Fast probability integration by three-parameter normal tail approximation. Struct Saf l(4) 269-276... 

If P x) is normalized, the integral in the denominator on the right is 1.) Comparing Equations 8.4 and 8.5 suggests that if C(x)m (x), called the absolute square or the square of the wavefunction, is a probability density, then Equation 8.4 will give the average value of x for the quantum mechanical system described by /. 

The value in the denominator is the probability of finding the particle somewhere or anywhere. The condition of normalization means that we choose this probability integral— the sum of probabilities of finding the particle at all positions— to be unity. Normalization is not implicit in the solution of the Schrodinger equation. Notice that when r fo is an eigenfunction of some Hamiltonian H, then the function T = 3r fo, for example, is just as much an eigenfunction of the Hamiltonian as r fo, and it has the same eigenvalue ... 

This is the familiar error function (Gaussian normal probability distribution), and the exact value when p = 1 is 0.84270073517 (this can be verified by using the Excel function ERF (p)). Remember to change variables so that the interval of integration is from -1 to 1. 

Equation (1-23) gives the probability of an event occurring within an arbitrary interval [a, b (Fig. 1-5). Equation (1-23) has been normalized by choosing the right premultiplying constant to make the integral over all space [—oo, oo] come out to 1.00. (see Problems) so the probability over any smaller interval [a, b] has a value not less than zero and not more than one. 

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