Gaussian and Poisson distributions

Gaussian and Poisson distributions are related in that they are extreme forms of the Binomial distribution. The binomial distribution describes the probability distribution for any number of discrete trials. A Gaussian distribution is therefore used when the probability of an event is large (this results in more symmetric bell-shaped curves), whereas a Poisson distribution is used when the probability is small (this results in asymmetric curves). The Lorentzian distribution represents... 

The likelihood of a particular event is generally described by the mean, median, and standard deviation (tr). The word generally is used as Lorentzian distributions do not have a mean, and hence a values, whereas Gaussian and Poisson distributions do. The median represents the middle of the range of values modeled. Note A mean is needed to derive average value exhibited by the distribution and a describes the likelihood from the mean value over which a particular event will occur, i.e. 68.3% occur within lmean value, 95.5% occur within 2a, and 99.7% occur within 3[Pg.294]

Both the binomial and Poisson distributions apply to discrete variables, whereas most of the random variables involved in experiments are continuous. In addition, the use of discrete distributions necessitates the use of long or infinite series for the calculation of such parameters as the mean and the standard deviation (see Eqs. 2.47, 2.48, 2.52, 2.53). It would be desirable, therefore, to have a pdf that applies to continuous variables. Such a distribution is the normal or Gaussian distribution. 

Bascom and Jensen [67], used an approach similar to that of Drzal and coworkers. Wimolkiatisak et al. [70] found that the fragmentation length data fitted both the Gaussian and Weibull distributions equally well. Fraser et al. [71 ] developed a computer model to simulate the stochastic fracture process and, together with the shear-lag analysis, described the shear transmission across the interface. Netravali et al. [72], used a Monte Carlo simulation of a Poisson-Weibull model for the fiber strength and flaw occurrence to calculate an effective interfacial shear strength X using the relationship ... 

We now consider probability theory, and its applications in stochastic simulation. First, we define some basic probabihstic concepts, and demonstrate how they may be used to model physical phenomena. Next, we derive some important probability distributions, in particular, the Gaussian (normal) and Poisson distributions. Following this is a treatment of stochastic calculus, with a particular focus upon Brownian dynamics. Monte Carlo methods are then presented, with apphcations in statistical physics, integration, and global minimization (simulated annealing). Finally, genetic optimization is discussed. This chapter serves as a prelude to the discussion of statistics and parameter estimation, in which the Monte Carlo method will prove highly usefiil in Bayesian analysis. 

What distribution does concern him It turns out to be the Gaussian, for the Poisson distribution as N becomes larger approaches more and more closely to the Gaussian. It may be shown analytically that, for large N,... 

The gaussian distribution is a good example of a case where the mean and standard derivation are good measures of the center of the distribution and its spread about the center . This is indicated by an inspection of Fig. 3-3, which shows that the mean gives the location of the central peak of the density, and the standard deviation is the distance from the mean where the density has fallen to e 112 = 0.607 its peak value. Another indication that the standard deviation is a good measure of spread in this case is that 68% of the probability under the density function is located within one standard deviation of the mean. A similar discussion can be given for the Poisson distribution. The details are left as an exercise. 

Exercise. Ordinary cumulants are adapted to the Gaussian distribution and factorial cumulants to the Poisson distribution. Other cumulants can be defined that are adapted to other distributions. For instance, define the %m by... 

Exercise. The property that the sum of two independent Gaussian variables is again Gaussian is not unique. Prove that the Lorentz and the Poisson distribution have a similar property. [Compare the Remark in 7.]... 

Remark. The white noise limit is not sufficiently defined by just saying rc 0. We have to construct a sequence of processes which in this limit reduce to Gaussian white noise. For that purpose take a long time interval (0, T) and a Poisson distribution of time points Ta in it with density v. To each Ta attach a random number ca they are independent and identically distributed, with zero mean. Consider the process... 

Examples of distributions are the binomial and the POISSON distribution and the GAUSSian (normal), y2-, t-, and -distributions. 

The variation that is observed in experimental results can take many different forms or distributions. We consider here three of the best known that can be expressed in relatively straightforward mathematical terms the binomial distribution, the Poisson distribution and the Gaussian, or normal, distribution. These are all forms of parametric statistics which are based on the idea that the data are spread in a specific manner. Ideally, this should be demonstrated before a statistical analysis is carried out, but this is not often done. 

In the Poisson and binomial distributions, the mean and variance are not independent quantities, and in the Poisson distribution they are equal. This is not an appropriate description of most measurements or observations, where the variance depends on the type of experiment. For example, a series of repeated weighings of an object will give an average value, but the spread of the observed values will depend on the quality and precision of the balance used. In other words, the mean and variance are independent quantities, and different two parameter statistical distribution functions are needed to describe these situations. The most celebrated such function is the Gaussian, or normal, distribution ... 

This is a continuous function for the experimental variables, which is used as a convenient mathematical idealisation to describe the distribution of finite numbers of results. The factor 1 /(ay/lji) is a constant such that the total area under the probability distribution curve is unity. The mean value is given by p and the variance by a2. The variance in the Gaussian distribution corresponds to the standard deviation s in Eqn. 8.3. Figure 8-3 illustrates the Gaussian distribution calculated with the same parameters used to obtain the Poisson distribution in Figure 8-2, i.e. a mean of 40 and a standard deviation of V40. It can be seen that the two distributions are similar, and that the Poisson distribution is very dosely approximated by the continuous Gaussian curve. 

Numerical calculations have been carried out for (i) ternary systems consisting of rods of two lengths x and x and a diluent with x = 1 (ii) solutions of polydisperse rods having a most probable distribution (iii) a Poisson distribution of rods in solution and (iv) various Gaussian distributions of rods in a diluent In all cases longer rods are preferentially partitioned into the nematic phase. For Xa = 2xb in case (i) the ratio of the concentration of either of the species in one of... 

The traditional statistics are not a proper way to evaluate the accuracy or precision of the results obtained on the basis of detecting only a few decay events. This is because the Gaussian approximation for the data distribution is not valid and the errors come from the statistical uncertainty, rather than from the imperfect measurements. To allow more rigorous treatment of low-level counting data, some authors updated the traditional approach by taking into account the inherent Poisson distribution [10,11]. 

We find that the Gaussian distribution is a more useful exemplar for time series than is the Poisson distribution used by Taylor. In Fig. 1 we apply Eq. (9) to one million computer-generated data points with Gaussian statistics. The far left dot in Fig. 1 contains all the data in the calculation of the aggregated mean and variance so that n = 1 in Eq. (9) the next point to the right in the figure... 

Generalized Unear mixed models (GLMMs) provide another type of extension of LME models aimed at non-Gaussian responses, such as binary and count data. In these models, conditional on the random effects, the responses are assumed independent and with distribution in the exponential family (e.g., binomial and Poisson) (8). As with NLME models, exact likelihood methods are not available for GLMMs because they do not allow closed form expressions for the marginal distribution of the responses. QuasUikelihood (9) and approximate likelihood methods have been proposed instead for these models. 

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