Probability Distributions and Mean Values

I EXERCISE 5.21 [ Write Mathematica entries to obtain the following integrals  

In this section, we discuss how to obtain certain average values using integration. There are several kinds of averages in common use. One type is the median, which is the value such that half of the set of values is greater than the median and half of the set is smaller than the median. The mode is the value that occurs most fie-quently in the set. We now discuss the calculation of a mean value by integration. The mean of a set of values is defined as 

There is another way to write the mean of a set of values if several of the members of the list are equal to each other. Let us arrange the members of our list so that the first M members of the list are all different from each other, and each of the other N — M members is equal to some member of the first subset. Let Ni be the total number of members of the entire list that are equal to x,, where x, is one of the distinct values in the first subset. Equation (5.59) can be rewritten 

We again use the standard notation of a capital Greek sigma (J ) for a siun, introduced in Eq. (5.17). The quantity p, is equal to Ni/N and is the fraction of the members of the entire list that are equal to x,. If we were to sample the entire list by choosing a member at random, the probability that this member would equal x, 

A set of probabilities that adds up to unity is said to be normalized. 

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After defining fundamental terms used in probability and introducing set notation for events, we consider probability theorems facilitating tlie calculation of the probabilities of complex events. Conditional probability and tlie concept of independence lead to Bayes theorem and tlie means it provides for revision of probabilities on tlie basis of additional evidence. Random variables, llicir probability distributions, and expected values provide tlie means... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

Suppose an interstate highway passes 1 km perpendicular distance from a nuclear power plant control room air intake on which 10 trucks/day pass carrying 10 tons bf chlorine each. Assume the probability of truck accident is constant at l.OE-8/mi, but if an accident occurs, the full cargo is released and the chlorine flashes to a gas. Assume that the winds are isotropically distributed with mean values of 5 mph and Pasquill "F" stability class. What is the probability of exceeding Regulatory Guide 1-78 criteria for chlorine of 45 mg/m (15 ppm). 

Figure 3 shows the joint distribution of V and CLtot (total clearance) for males and females as calculated by the model (C). Volume and clearance are distributed around mean values (center of the ellipse) and they are slightly correlated to each other. The 95% contour line of their joint probability of occurrence is shown as ellipses for male and female. 

The sign test to compare two treatments. We assume that there are several independent pairs of observations on the two treatments. The hypothesis to be tested states that each difference has a probability distribution having mean equal to zero. For each difference the algebraic sign is noted and then the number of times the less frequent sign is considered as the test statistic. There are speciahzed tables for the critical value of this quantity once a level of significance is chosen. 

Treating AFg as a Hamiltonian of the fluctuating variable the probability to find h q) with any particular value is proportional to exp[— which in our approximation is a Gaussian where all the different q modes are independent. Using the relationship between the probability distribution and the mean-square value of a fluctuating quantity (see Chapter 1), we have... 

Table 1. Probability distributions and parameters (i.e., means and standard deviations) of the uncertain variables xi, X2, x, and X4 of the cracked plate model of Section 4.2 for the four case studies considered (i.e.. Cases 0,1,2 and 3) the last row reports the values of the corresponding exact (i.e., analytically computed) failure probabilities, P(F) (Gille 1999).

Equation (30) can be solved if Fy is sampled at each time step in some specified way. If Fy is assumed to be a Gaussian Markov process it follows from Doob s theorem that Kj(t) is an exponential function of time. Then only two parameters need be specified before Fy can be sampled from the Gaussian two-time probability distribution and these are the mean square value (Fy) and the correlation time of Fy, say xy. Equation (30) then forms a set of coupled stochastic differential equations that can be solved by methods similar to those already mentioned. 

For the discussion of the chaotic behavior statistical methods will be used. The relative cumulative frequency or the probability, respectively, is shown in Figure 7 for four velocity classes, see Table 3. From these data the relative frequency or probability density, respectively, is obtained, see Figure 8. It turns out that the frequency distribution is completely non-Gaussian, and the range characterizing the statistical dispersion is increasing with the relative velocity of the impact, while midrange point and mean value coincide fairly well, see Table 4. 

There are many different ways to treat mathematically uncertainly, but the most common approach used is the probability analysis. It consists in assuming that each uncertain parameter is treated as a random variable characterised by standard probability distribution. This means that structural problems must be solved by knowing the multi-dimensional Joint Probability Density Function of all involved parameters. Nevertheless, this approach may offer serious analytical and numerical difficulties. It must also be noticed that it presents some conceptual limitations the complete uncertainty parameters stochastic characterization presents a fundamental limitation related to the difficulty/impossibility of a complete statistical analysis. The approach cannot be considered economical or practical in many real situations, characterized by the absence of sufficient statistical data. In such cases, a commonly used simplification is assuming that all variables have independent normal or lognormal probability distributions, as an application of the limit central theorem which anyway does not overcome the previous problem. On the other hand the approach is quite usual in real situations where it is only possible to estimate the mean and variance of each uncertainty parameter it being not possible to have more information about their real probabilistic distribution. The case is treated assuming that all uncertainty parameters, collected in the vector d, are characterised by a nominal mean value iJ-dj and a correlation =. In this specific... 

The Poisson white noise E (t) consists of a sequence of delta peaks at random points in time. The average time difference between two peaks is controlled by and the amplitude w of the peaks is distributed according to a probability density with mean value w. The process y (t) is bounded from below, (t) X w. 

In order to determine the First Hitting Time (FHT) distribution of the individual and mean value, we set histograms and performed tests for a presumed type of distribution. The expected types of probability density distribution such as Gamma... 

One measure of the width of the peak is with the variance. The variance of a discrete distribution is denoted by and is defined as the mean of the square of the deviation of the probabilities from the mean value ... 

Interestingly, the waiting-time probability distribution, and hence the corresponding mean value, depend on the current state of the system. 

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