Probability distribution average

Using the f value, let us define the time-dependent quality 2 [f (f)/f (0) - 1] and three intermediate folding times fi/4, t]/2, and 3/4, describing its evolution, as well as the corresponding sequence-averaged probability distribution functions Wi/4, Wi/2, and W3/4. The distribution of folding times averaged over 1000 different sequences of 128-unit HP copolymers with random, random-block, and protein-like statistics are shown in Fig. 26. 

The average probability distribution of the local polarization p is defined as... 

We have already seen in Eq. (16) that the usual correlation function is the thermal correlation of the pure problem and has no signature of the dborder. Of course, the disordered averaged probability distribution P r,N) = [Z r,N)/ f drZ(r,N)]. is also of importance. 

Hsieh, T., Okrent, D., and Apostolakis, O. E. On the Average Probability Distribution of Peak Ground Acceleration in the U.S. Continent Due to Strong Earthquakes. UCLA-ENG-75 16, University of California at Los Angeles, March 1976. 

These probabilities were obtained by averaging over system microstates, so we call them an average probability distribution. When we use these probabilities to calculate an average molecular quantity we will actually have a double average an average over molecular states using a probability distribution that was itself obtained by averaging... 

In either case, first-order or continuous, it is usefiil to consider the probability distribution function for variables averaged over a spatial block of side L this may be the complete simulation box (in which case we... 

We propose to describe the distribution of the number of fronts crossing x by the Poisson distribution function, discussed in Sec. 1.9. This probability distribution function describes the probability P(F) of a specific number of fronts F in terms of that number and the average number F as follows [Eq. (1.38)] ... 

The statistical average of a variable described by a probability distribution... 

To obtain thermodynamic averages over a canonical ensemble, which is characterized by the macroscopic variables (N, V, T), it is necessary to know the probability of finding the system at each and every point (= state) in phase space. This probability distribution, p(r, p), is given by the Boltzmann distribution function. 

An important question is whether one can rigorously express such an average without referring explicitly to the solvent degrees of freedom. In other words. Is it possible to avoid explicit reference to the solvent in the mathematical description of the molecular system and still obtain rigorously correct properties The answer to this question is yes. A reduced probability distribution P(X) that depends only on the solute configuration can be defined as... 

The reduced probability distribution does not depend explicitly on the solvent coordinates Y, although it incorporates the average influence of the solvent on the solute. The operation symbolized by Eq. (4) is commonly described by saying that the solvent coordinates Y have been integrated out. In a system at temperature T, the reduced probability has the form... 

Because the datay are random, the statistics based on y, S(y), are also random. For all possible data y (usually simulated) that can be predicted from H, calculate p(S(ysim) H), the probability distribution of the statistic S on simulated data y ii given the truth of the hypothesis H. If H is the statement that 6 = 0, then y i might be generated by averaging samples of size N (a characteristic of the actual data) with variance G- = G- (yacmai) (yet another characteristic of the data). 

In general a macroscopic observable can be calculated as an average over a corresponding microscopic quantity weighted by the Boltzman probability distribution as in eq. (16.5). 

If chains are long such that the initiation and termination reactions have a negligible effect on the average sequence distribution, then according to the terminal model, PAA, the probability that a chain ending in monomer unit MA adds another unit MA, is given by eq. 22 8... 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

There is thus assumed to be a one-to-one correspondence between the most probable distribution and the thermodynamic state. The equilibrium ensemble corresponding to any given thermodynamic state is then used to compute averages over the ensemble of other (not necessarily thermodynamic) properties of the systems represented in the ensemble. The first step in developing this theory is thus a suitable definition of the probability of a distribution in a collection of systems. In classical statistics we are familiar with the fact that the logarithm of the probability of a distribution w[n is — J(n) w n) In w n, and that the classical expression for entropy in the ensemble is20... 

It will be assumed for the moment that the non-bonded atoms will pass each other at the distance Tg (equal to that found in a Westheimer-Mayer calculation) if the carbon-hydrogen oscillator happens to be in its average position and otherwise at the distance r = Vg + where is a mass-sensitive displacement governed by the probability distribution function (1). The potential-energy threshold felt is assumed to have the value E 0) when = 0 and otherwise to be a function E(Xja) which depends on the variation of the non-bonded potential V with... 

The angular-dependent adiabatic potential energy curves of these complexes obtained by averaging over the intermolecular distance coordinate at each orientation and the corresponding probability distributions for the bound intermolecular vibrational levels calculated by McCoy and co-workers provide valuable insights into the geometries of the complexes associated with the observed transitions. The He - - IC1(X, v" = 0) and He + 1C1(B, v = 3) adiabatic potentials are shown in Fig. 3 [39]. The abscissa represents the angle, 9,... 

The viscosity average molecular weight depends on the nature of the intrinsic viscosity-molecular weight relationship in each particular case, as represented by the exponent a of the empirical relationship (52), or (55). However, it is not very sensitive to the value of a over the range of concern. For polymers having the most probable distribution to be discussed in the next chapter, it may be shown, for example, that... 

Distribution curves calculated for several values of / are shown in Fig. 56. Values of p have been adjusted to give the same number average (see Eq. 23), which also locates the maxima in the curves very nearly at the same abscissa value. The sharpening of the curves with increase in / is evident. The curve for /= 1, corresponding to the most probable distribution, is included for comparison. Even for /=2, which represents the linear polymer prepared by condensing... 

The portion of the polymer consisting of molecules terminated by transfer will conform to the most probable distribution, its average degree of polymerization being... 

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