Probability function independent

A parallel system is one lliat fails to operate only if all its components fail to operate. If R, is tlie reliability of the i component, llien (1-Ri) is tlie probability tliat tlie i component fails i = 1,. .., n. Assuming lliat all n components function independently, tlie probability tliat all n components fail is (1-Ri)(l-R2)...(l-Rn). Suiitracting lliis product from unity yields the following formula for Rp, tlie reliability of a parallel system. ... 

In our discussion of the electron density in Chapter 5, I mentioned the density functions pi(xi) and p2(xi,X2). I have used the composite space-spin variable X to include both the spatial variables r and the spin variable s. These density functions have a probabilistic interpretation pi(xi)dridii gives the chance of finding an electron in the element dri d i of space and spin, whilst P2(X], X2) dt] d i dt2 di2 gives the chance of finding simultaneously electron 1 in dri dii and electron 2 in dr2di2- The two-electron density function gives information as to how the motion of any pair of electrons is correlated. For independent particles, these probabilities are independent and so we would expect... 

Great simplification is achieved by introducing the hypothesis of independent reaction times (IRT) that the pairwise reaction times evolve independendy of any other reactions. While the fundamental justification of IRT may not be immediately obvious, one notices its similarity with the molecular pair model of homogeneous diffusion-mediated reactions (Noyes, 1961 Green, 1984). The usefulness of the IRT model depends on the availability of a suitable reaction probability function W(r, a t). For a pair of neutral particles undergoing fully diffusion-con-trolled reactions, Wis given by (a/r) erfc[(r - a)/2(D t)1/2] where If is the mutual diffusion coefficient and erfc is the complement of the error function. 

In Eq. (42) F may be considered as a random variable in the coordinates of the N2 defects. The distribution law is that every defect can take up any one of the sites on its sublattice with equal probability and independently of the positions of the other defects. The correlation between the positions of the defects implied by the original distribution law in Eq. (41) has been removed by introducing the h functions. 

In words, the right-hand side is the probability that the random variable U (x, t) falls between the sample space values V) and V) + dV) for different realizations of the turbulent flow.5 In a homogeneous flow, this probability is independent of x, and thus we can write the one-point PDF as only a function of the sample space variable and time /(Vi i ). 

Note that its contribution to the probability function makes certain limiting behaviors on A Aii(r) intuitively obvious. For instance, the function should go to zero very rapidly when r becomes less than the sum of the van der Waals radii of A and B. In addition, at very large r, the function should be independent of r in homogeneous media, like fluids,... 

The oxygen affinity of the derivative was shown to be about half that of unmodified hemoglobin under similar conditions, but a degree of cooperativity was preserved. Equilibrium and kinetic ligand-binding studies on this derivative have been interpreted (62) to show a perturbed R state. It is believed that although the reaction is between the two ft-chains, ocP-dimers function independently, probably through a flexible connection. 

This result for P2(ri,ra) can only be true in the limit that the motion of electron 1 and hence its probability of being at a certain point in space is totally independent of the motions of electron 2, i.e. there are no forces acting between the two electrons and their motions are totally uncorrelated. The best that can be done in a model which employs an expression of the form of equation (19) for the pair probability function is to determine the m (which in turn determine the one-particle probability function) in such a way that the electron it describes experiences the average field of the other electrons, a situation which is attained in the Hartree-Pock limit of the orbital model. 

We will find it convenient to consider for the moment the special case of perfectly reflecting walls. In the long time-scale limit A a /D, the average propagator for fully restricted diffusion has a simple relationship to the pore geometry. This requirement on A, also known as the pore equilibration condition, implies that the time is sufficiently long that most molecules have collided with the walls. Under this condition the conditional probabilities are independent of starting position so that P(r, t I r, 0) reduces to p(r ), the pore molecular density function. 

The ancestral ribosome is thought to have contained more proteins than are now found in the archaeal and bacterial ribosomes, and may have lost these extra r-proteins by a streamlining process [8,10,147]. During this process, the function of various r-proteins may have been combined and the number of r-proteins in the ribosome thereby reduced. Since this streamlining probably occurred independently in the archaea and bacteria, different proteins may have resulted from the combination of these functional components [169]. This hypothesis would be supported by the observation that some r-proteins found in archaea do not seem to have an equivalent in bacteria. 

When approximate solutions are extended to heavier atoms and to molecules, it is necessary to be content with a rather lower accuracy (e.g, 1 to 2%). The most fruitful approximation method is that of Hartree (1928), later justified and refined by Slater (1930) and Fock (1930). This is suggested by lirst neglecting the electron repulsion and observing that the probability function P (1, 2,. .. N) must then be approximated by the product (1) Pg(2). .. Pj (N), for then the probability of any configuration of the electrons is a product of the probabilities of N independent events. Since P = Pf, this means the many-electron wave function must also be a product... 

In other words, pjk(n), the n-step transition probability function, is the conditional probability of occupying Sk at the nth step, given that the system initially occupied Sj. pjk(n), termed also higher transition probability, extends the one-step transition probability pjk(l) = Pjk and gives an answer to question 2 in 2.1-2. Note also that the function given by Eq.(2-26) is independent of t, since we are concerned in homogeneous transition probabilities. 

According to the Classical Nucleation Theory, the repetitive formation of nuclei in a metastable liquid can be considered as a sequence of independent events and the distribution of metastable lifetimes shows an exponential decrease (see also Takahashi et This implies that the density probability function f(t) of the nucleation event is ... 

A cumulative probabihty distribution function characterizes a set of outcomes between an upper and a lower bound. For a discrete distribution function, the associated distribution is usually denoted as F(a upper bounds, respectively. The new function F is determined by summing the probability of independent outcomes that result in x between a and... 

The binomial distribution describes the probability of x successes in n trials when only success or failure occurs. This distribution assumes that each trial is independent and that the probability of a success in each trial is constant. This probability is denoted p. The probability function for a binomial distribution [2] is... 

However, equation (24) assumes that the probability is independent of the trajectory of the projectile. In order to incorporate interference effects, one introduces into equation (16) the electronic wave function Up). Thus, projecting the final wave function onto a final state f) and repeating the Schiff procedure, one obtains... 

Consider a uniform continuous cash flow, A, which begins at time m and continues for an uncertain duration t. Assume that m and t are statistically independent random variables with known probability functions f m) and f t). It may be shown that... 

To illustrate, consider the problem summarized in Table 3. There are three risky, independent, end-of-year cash flows the means and variances of their respective probability functions are given in columns (2) and (3) of Table 3. Project life, N, is also a random variable, with probability mass function as shown in columns (6) and (7) of the table. A 10% discount rate is assumed. Determination of the expected present worth, based on Eq. 17, is summarized in the table. Here, fXp = — 82.64. The variance of present worth, 0-2, based on Eqs. 18 and 19, may be shown to be... 

The assumption that each turbulent eddy spends the same amount of time at the interface is unrealistic. Modified penetration models, such as Danckwerts surface renewal theory model, allow each eddy to have an independent, variable interface contact time based on a statistical probability function. It uses a fractional renewal s to account for the rate of surface renewal, in which case ... 

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