Uniform probability distribution

Application of this equation to the probability distributions given in Table 40.6 shows that H for the less precise method is larger than for the more precise method. Uniform distributions represent the highest form of uncertainty and disorder. Therefore, they have the largest entropy. 

The basic idea of importance sampling can be illustrated simply in the example of the transformation from 0 to 1 along A, as described above. In lieu of sampling from the true probability distribution, P A), we design simulations in which A is sampled according to P A). The latter probability should be chosen so that it is more uniform than P A). The relation between the two probabilities may then be expressed as follows ... 

Cai, X. D., E. E. O Brien, and F. Ladeinde (1996). Uniform mean scalar gradient in grid turbulence Asymptotic probability distribution of a passive scalar. Physics of Fluids 8, 2555-2557. 

Models are often best understood relative to the situation they are designed to describe if their constitutive variables are allowed to fluctuate statistically in a realistic way. Once a variable has been assigned a suitable density of probability distribution and the parameters of this distribution have been chosen, the fluctuations can be conveniently produced by using random deviates from statistical tables. A random deviate is a particular value of a standard random variable. Many elementary books in statistics contain tables of deviates from uniform, normal, exponential,. .. distributions. Many high-level computation-oriented programming languages (e.g., MatLab) and spreadsheets, such as Microsoft Excel, also contain random number generators. The book by Press et al. (1986) contains software that produces random deviates for the most commonly used probability distributions. 

In the Monte Carlo analysis the tolerance distribution has a major effect on the results. It is best to know the distribution of your parts before you trust the results of the Monte Carlo analysis. You can use the PSpice .Distribution statement to define your own probability distributions. If you do not know the distribution of your parts, the Gaussian distribution is a better representation of parts than the uniform distribution. 

In previous section, ensembles with well-separated constants appear. We represented them by a log-uniform distribution in a sufficiently big interval log ke[a, jS], but we were not interested in most of probability distribution properties, and did not use them. The only property we really used is if fcj >fcy, then ki/kj 1 (with probability close to one). It means that we can assume that ki/kj a for any preassigned value of a that does not depend on k values. One can interpret this property as an asymptotic one for a — co,p- co. 

Exercise. Prove that the characteristic function of any probability distribution is uniformly continuous on the real k-axis. 

Exercise. A point lies on a circle at a random position with uniform distribution along the circle. When viewed by an eccentric observer, what is the probability distribution of its azimuth ... 

Detritus from a paleolandscape contains information on the paleorelief (in fact, the paleohypsometry), of its former catchment, through its probability distribution of cooling ages, provided the paleoerosion rate can be estimated, and some assumption about the spatial uniformity of erosion (and bedrock abundance of the thermochronometric mineral) can be made. The former could potentially be estimated from mineral-pair methods, such as using coupled apatite He and apatite FT ages from the same clast, or from AERs of bedrock samples that record the erosion rate through the time interval in question. In general, the latter constraint must be assumed, however. 

Selecting the points for crossover and mutation according to a probability distribution, either uniform or skewed towards points at which the optimized function takes high values (the latter being a probabilistic expression of the survival-of-the-fittest principle). 

The probability distribution of the values of any proportion or any ratio between proportions on any prescribed interval (such as the uniform distribution, exponential distribution, or log-uniform distribution). 

The bubble size detected by the probe is subject to a probability distribution [28,29] and only the mean statistical value of the bubbles is given by J32. The analysis of the bubble size distribution helps us understand the uniformity of the bubble sizes and its axial evolution. Fig. 11 shows the axial change of the bubble size distribution at r/R = 0 and r/R = 0.97, respectively. Below the internal, the peak... 

The basic assumption in statistical theories is that the initially prepared state, in an indirect (true or apparent) unimolecular reaction A (E) —> products, prior to reaction has relaxed (via IVR) such that any distribution of the energy E over the internal degrees of freedom occurs with the same probability. This is illustrated in Fig. 7.3.1, where we have shown a constant energy surface in the phase space of a molecule. Note that the assumption is equivalent to the basic equal a priori probabilities postulate of statistical mechanics, for a microcanonical ensemble where every state within a narrow energy range is populated with the same probability. This uniform population of states describes the system regardless of where it is on the potential energy surface associated with the reaction. 

Each phase is characterized by a mean duration D and a variability V. As soon as the prescribed duration of a given phase is reached, the transition to the next phase of the cell cycle occurs. The time at which the transition takes place varies in a random manner according to a distribution of durations of cell cycle phases. In the case of a uniform probability distribution, the duration varies in the interval [D(l - V), D( 1 + V)]. 

All the above results have been obtained for the case where the durations of the various cell cycle phases obey a probability distribution centered around the mean duration D, with a range of variation extending uniformly from D — V to D + V. Similar results are obtained when assuming that the probability distribution obeys a lognormal distribution centered around the same mean value [33]. 

When constructing input distributions for an uncertainty analysis, it is often useful to present the range of values in terms of a standard probability distribution. It is important that the selected distribution be matched to the range and moments of any available data. In some cases, it is appropriate to simply use the raw data or a custom distribution. Other more commonly used standard probability distributions include the normal distribution, the lognormal distribution, the uniform distribution, the log-uniform distribution and the triangular distribution. For the case-study presented below, we use lognormal distributions. 

Noncompartmental models were introduced as models that allow for transport of material through regions of the body that are not necessarily well mixed or of uniform concentration [248]. For substances that are transported relatively slowly to their site of degradation, transformation, or excretion, so that the rate of diffusion limits their rate of removal from the system, the noncompartmental model may involve diffusion or other random walk processes, leading to the solution in terms of the partial differential equation of diffusion or in terms of probability distributions. A number of noncompartmental models deal with plasma time-concentration curves that are best described by power functions of time. 

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