Random range

Despite certain intrinsic statistical disadvantages the linear least-square procedures can sometimes provide satisfactory rlf r2 estimates when the copolymerization experiment is properly designed. Instead of conducting all the measurements over the random range of the initial monomer feed composition, one should carry them out only at the two following values ... 

The copolymerizations should not be carried out over a random range of feed compositions, bul the available effort should be devoted to replications of copolymerizations at two monomer feeds /, and f[ given by... 

The particles in liquids, just as in gases, are moving in random directions with a random range of speeds. Some of them must be moving upwards. Some of these will reach the surface. There they experience a downward attraction, but some of them will be moving fast enough to escape. The liquid evaporates. 

A random number (between 0 and 1) is picked, and the associated value of gross reservoir thickness (T) is read from within the range described by the above distribution. The value of T close to the mean will be randomly sampled more frequently than those values away from the mean. The same process is repeated (using a different random number) for the net-to-gross ratio (N/G). The two values are multiplied to obtain one value of net sand thickness. This is repeated some 1,000-10,000 times, with each outcome being equally likely. The outcomes are used to generate a distribution of values of net sand thickness. This can be performed simultaneously for more than two variables. 

In the limit that the number of effective particles along the polymer diverges but the contour length and chain dimensions are held constant, one obtains the Edwards model of a polymer solution [9, 30]. Polymers are represented by random walks that interact via zero-ranged binary interactions of strength v. The partition frmction of an isolated chain is given by... 

The random-bond heteropolymer is described by a Hamiltonian similar to (C2.5.A3) except that the short-range two-body tenn v.j is taken to be random with a Gaussian distribution. In this case a tliree-body tenn with a positive value of cu is needed to describe the collapsed phase. The Hamiltonian is... 

Having generated a trial conformation it must be decided whether to accept it or not. To d this a random number is generated in the range 0-1 and compared with the ratio of tf Rosenbluth weights for the trial conformation (W trial) and the old conformation (W/oid The new chain is then accepted using the following criterion ... 

The energies may be random within some fixed range. Random-number generators use this property intentionally. 

It will be convenient to deal first with the distribution aspect of the problem. One of the clearest ways in which to represent the distribution of sizes is by means of a histogram. Suppose that the diameters of SOO small spherical particles, forming a random sample of a powder, have been measured and that they range from 2-7 to 5-3 pm. Let the range be divided into thirteen class intervals 2-7 to 2-9 pm, 2-9 to 3-1 pm, etc., and the number of particles within each class noted (Table 1.5). A histogram may then be drawn in which the number of particles with diameters within any given range is plotted as if they all had the diameter of the middle of the... 

Uncertainty expresses the range of possible values that a measurement or result might reasonably be expected to have. Note that this definition of uncertainty is not the same as that for precision. The precision of an analysis, whether reported as a range or a standard deviation, is calculated from experimental data and provides an estimation of indeterminate error affecting measurements. Uncertainty accounts for all errors, both determinate and indeterminate, that might affect our result. Although we always try to correct determinate errors, the correction itself is subject to random effects or indeterminate errors. 

Earlier we noted that 68.26% of a normally distributed population is found within the range of p, lo. Stating this another way, there is a 68.26% probability that a member selected at random from a normally distributed population will have a value in the interval of p, lo. In general, we can write... 

Range of results around a mean value that could be explained by random error. 

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