Distribution of a population

To predict the properties of a population on the basis of a sample, it is necessary to know something about the population s expected distribution around its central value. The distribution of a population can be represented by plotting the frequency of occurrence of individual values as a function of the values themselves. Such plots are called prohahility distrihutions. Unfortunately, we are rarely able to calculate the exact probability distribution for a chemical system. In fact, the probability distribution can take any shape, depending on the nature of the chemical system being investigated. Fortunately many chemical systems display one of several common probability distributions. Two of these distributions, the binomial distribution and the normal distribution, are discussed next. 

It is possible to assess the fraction of defects in a crystal that is associated using the approximate interaction energy calculated above using the Boltzmann law. This gives the information about the distribution of a population of defects between two energy states. The fraction / of the population in the upper energy state, when the energy difference between the states is AE, is... 

Figure 7.4 Typical frequency distribution of a population response to an equivalent dose of a biologically active agent. This type of response represents the variability that occurs within biological systems and is the basis for the concept of dose response in pharmacology and toxicology. This figure demonstrates that within any population, both hyporeactive and hyperreactive individuals can be expected to exist and must be addressed in a risk assessment.

The distribution of a population s property can be introduced mathematically by the repartition function of a random variable. It is well known that the repartition function of a random variable X gives the probability of a property or event when it is smaller than or equal to the current value x. Indeed, the function that characterizes the density of probability of a random variable (X) gives current values between X and x -I- dx. This function is, in fact, the derivative of the repartition function (as indirectly shown here above by relation (5.16)). It is important to make sure that, for the characterization of a continuous random variable, the distribution function meets all the requirements. Among the numerous existing distribution functions, the normal distribution (N), the chi distribution (y ), the Student distribution (t) and the Fischer distribution are the most frequently used for statistical calculations. These different functions will be explained in the paragraphs below. 

In Section 4, competition between two populations is analyzed. Again, the equations can be reduced to a system that can be directly compared to the systems derived in Chapters I and 2. Section 5 explores the evolution in time of the population average length, surface area, and volume in Section 6 we formulate the conservation principle, which played such a crucial role in earlier chapters. The steady-state size distribution of a population is determined in Section 7. Our findings are summarized in a discussion section, where a comparison is made between the conclusions derived from the size-structured model and the unstructured models considered in Chapters 1 and 2. 

Gaussian Normally distributed or shaped like the familiar bell curve. Gaussian distribution A normal distribution of a population that is described by approximations of the Poisson distribution. 

The production of a nonequilibrium distribution of a population of magnetic or HFS sublevels in the ground state has foimd widespread application in high-precision atomic clocks and ultrasensitive magnetometers, and lately in noble-gas-based magnetic imaging. Let us very briefly consider these applications. 

To summarize this section we have developed a simple dynamical model for a genetic algorithm that predicts the time evolution of the fitness distribution of a population. As basis for its prediction, it uses the parent-child fitness correlation of the genetic operators with respect to the specific fitness function. What has been demonstrated here for the simplest fitness function, a random field paramagnet, has also been generalized and applied to more rugged landscapes in [47], e.g., the spin glass motivated NK model. 

A simple decision-making problem is I measure variable x of a population A and the same variable xof a population B. I get (slightly) different results. Is there areal difference between populations A and B based on the difference in measurements, or am I only seeing different parts of the distributions of identical populations ... 

In attempting to reach decisions, it is useful to make assumptions or guesses about the populations involved. Such assumptions, which may or may not be true, are called statistical hypotheses and in general are statements about the probability distributions of the populations. A common procedure is to set up a null hypothesis, denoted by which states that there is no significant difference between two sets of data or that a variable exerts no significant effect. Any hypothesis which differs from a null hypothesis is called an alternative hypothesis, denoted by Tfj. 

Binomial Distribution The binomial distribution describes a population in which the values are the number of times a particular outcome occurs during a fixed number of trials. Mathematically, the binomial distribution is given as... 

The binomial distribution describes a population whose members have only certain, discrete values. A good example of a population obeying the binomial distribution is the sampling of homogeneous materials. As shown in Example 4.10, the binomial distribution can be used to calculate the probability of finding a particular isotope in a molecule. 

If we randomly select a single member from a population, what will be its most likely value This is an important question, and, in one form or another, it is the fundamental problem for any analysis. One of the most important features of a population s probability distribution is that it provides a way to answer this question. 

In Section 4D.2 we introduced two probability distributions commonly encountered when studying populations. The construction of confidence intervals for a normally distributed population was the subject of Section 4D.3. We have yet to address, however, how we can identify the probability distribution for a given population. In Examples 4.11-4.14 we assumed that the amount of aspirin in analgesic tablets is normally distributed. We are justified in asking how this can be determined without analyzing every member of the population. When we cannot study the whole population, or when we cannot predict the mathematical form of a population s probability distribution, we must deduce the distribution from a limited sampling of its members. 

The nature of the surrounding population. The distribution of tlie population indoors varies depending on tlie lime of day and the season, tlie overall healtli of the population (senior citizens, infirm, etc.), and tlie type of clothing being worn (cotton, wool, polyester, etc.) by tlie personnel exposed to a possible heat radiation. 

If the probabilities do not remain constant over the trials and if there are k (rather than two) possible outcomes of each trial, the hypergeometric distribution applies. For a sample of size N of a population of size T, where... 

In analytical chemistry one of the most common statistical terms employed is the standard deviation of a population of observations. This is also called the root mean square deviation as it is the square root of the mean of the sum of the squares of the differences between the values and the mean of those values (this is expressed mathematically below) and is of particular value in connection with the normal distribution. 

The required distribution of initial populations ntu can be obtained in the following manner (32). Let us consider a system with Ed mi = 20 kcal/ mole and Ed max = 45 kcal/mole. Assuming that kd = 1013 sec-1 and x = 1, we can calculate theoretical desorption rates dnai/dt for Ed = 20, 21, 22,..., 45 kcal/mole as a function of nBOi. With increasing temperature, 25 values of dnjdt are measured at temperatures corresponding to Ed of 20, 21, 22,. . ., 45 kcal/mole. Since the total desorption rate at any moment must be equal to the sum of the individual desorption processes, we obtain 25 linear equations. Their solution permits the computation of the initial populations of the surface sites in the energy spectrum considered, i.e. the function n,oi(Edi). From the form of this function, desorption processes can be determined which exhibit a substantial effect on the experimental desorption curve. 

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

The Log-Probit Model. The log-probit model has been utilized widely in the risk assessment literature, although it has no physiological justification. It was first proposed by Mantel and Bryan, and has been found to provide a good fit with a considerable amount of empirical data (10). The model rests on the assumption that the susceptibility of a population or organisms to a carcinogen has a lognormal distribution with respect to dose, i.e., the logarithm of the dose will produce a positive response if normally distributed. The functional form of the model is ... 

Population Balance Approach. The use of mass and energy balances alone to model polymer reactors is inadequate to describe many cases of interest. Examples are suspension and emulsion polymerizations where drop size or particle distribution may be of interest. In such cases, an accounting for the change in number of droplets or particles of a given size range is often required. This is an example of a population balance. 

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