Sample distribution

Torrie G M and Valleau J P 1977 Nonphysical sampling distributions In Monte Carlo free energy estimation umbrella sampling J. Comput. Phys. 23 187-99... 

Sample Distributions and the Central Limit Theorem Let s return to the problem of determining a penny s mass to explore the relationship between a population s distribution and the distribution of samples drawn from that population. The data shown in Tables 4.1 and 4.10 are insufficient for our purpose because they are not large enough to give a useful picture of their respective probability distributions. A better picture of the probability distribution requires a larger sample, such as that shown in Table 4.12, for which X is 3.095 and is 0.0012. 

Determining the area under the normal cuiwe is a very tedious procedure. However, by standardizing a random variable that is normally distributed, it is possible to relate all normally distributed random variables to one table. The standardization is defined by the identity z = (x — l)/<7, where z is called the unit normal. Further, it is possible to standardize the sampling distribution of averages x by the identity = (x-[l)/ G/Vn). 

The sampling distribution of count data can be charac terized through probabihty distributions. In many cases, count data are appropriately interpreted through their corresponding distributions. However, in other situations analysis is greatly facilitated through distributions which have been developed for measurement data. Examples of each will be illustrated in the following subsections. 

Consider the hypothesis Ii = [Lo- If, iri fact, the hypothesis is correct, i.e., Ii = [Lo (under the condition Of = o ), then the sampling distribution of x — x is predictable through the t distribution. The obseiwed sample values then can be compared with the corresponding t distribution. If the sample values are reasonably close (as reflectedthrough the Ot level), that is, X andxg are not Too different from each other on the basis of the t distribution, the null hypothesis would be accepted. Conversely, if they deviate from each other too much and the deviation is therefore not ascribable to chance, the conjecture would be questioned and the null hypothesis rejected. 

I. Under the null hypothesis, it is assumed that the respective two samples have come from populations with equal proportions pi = po. Under this hypothesis, the sampling distribution of the corresponding Z statistic is known. On the basis of the observed data, if the resultant sample value of Z represents an unusual outcome, that is, if it falls within the critical region, this would cast doubt on the assumption of equal proportions. Therefore, it will have been demonstrated statistically that the population proportions are in fact not equal. The various hypotheses can be stated ... 

As the parent of actinium in this series it was named protoactinium, shortened in 1949 to protactinium. Because of its low natural abundance its chemistry was obscure until 1960 when A. G. Maddock and co-workers at the UK Atomic Energy Authority worked up about 130g from 60 tons of sludge which had accumulated during the extraction of uranium from UO2 ores. It is from this sample, distributed to numerous laboratories throughout the world, that the bulk of our knowledge of the element s chemistry was gleaned. 

FIGURE 11.23 Power analysis.The desired difference is >2 standard deviation units (X, - / = 8). The sample distribution in panel a is wide and only 67% of the distribution values are > 8. Therefore, with an experimental design that yields the sample distribution shown in panel a will have a power of 67% to attain the desired endpoint. In contrast, the sample distribution shown in panel b is much less broad and 97% of the area under the distribution curve is >8. Therefore, an experimental design yielding the sample distribution shown in panel B will gave a much higher power (97%) to attain the desired end point. One way to decrease the broadness of sample distributions is to increase the sample size. 

The molten salt standard program was initiated at Rensselaer Polytechnic Institute (RPI) in 1973 because of difficulties being encountered with accuracy estimates in the NBS-NSRDS molten salt series. The density, surface tension, electrical conductivity, and viscosity of KNO3 and NaCl were measured by seven laboratories over the world using samples distributed by RPI. The data from these round-robin measurements of raw properties were submitted to RPI for evaluation. Their recommendations are summarized in Table 2. 

It is useful to factor out Ca (f) and solve the differential equation in terms of just C[)(t). This can be done by taking into account the mass balance, which requires that the total amount of sample be preserved, and be distributed between the donor and the acceptor compartments (disregarding the membrane for now). At t 0, all the solute is in the donor compartment, which amounts to VpCp 0) moles. At time t, the sample distributes between two compartments ... 

When membrane retention of the solute needs to be considered, one can derive the appropriate permeability equations along the lines described in the preceding section Eqs. (7.1)—(7.3) apply (with P designated as the effective permeability, Pe). However, the mass balance would need to include the membrane compartment, in addition to the donor and acceptor compartments. At time t, the sample distributes (mol amounts) between three compartments ... 

The sampling distribution of count data can be characterized through probability distributions. In many cases, count data are appropriately... 

Table number Designated symbol Variable Sampling distribution of... 

Torrie, G. M. Valleau, J. R, Nonphysical sampling distributions in Monte Carlo free energy estimation Umbrella sampling, J. Comput. Phys. 1977, 23, 187-199... 

When calculating free energies, one generates, either by molecular dynamics or MC, configuration space samples distributed according to a probability distribution function (e.g., the Boltzmann distribution in the case of the Helmholtz free energy). 

To handle broken ergodicity in the calculation of thermodynamic averages (i.e., time-independent averages of the type (A) oc / /I (x) exp(— ft77(x))dx) a host of methods have been devised [2]. In a subset of these, the Boltzmann distribution is altered and replaced with a more delocalized one, w(E). One is thus able to generate (faster) samples distributed according to w and subsequently, one can use importance... 

As one of the possible ways to alter the sampling distribution in a manner that is conducive to enhanced sampling, we present a strategy based on probability distributions that arise in a generalization of statistical mechanics proposed by Tsallis [31]. In this... 

---