Distribution probabilities

The probability distribution of a randoni variable concerns tlie distribution of probability over tlie range of tlie random variable. The distribution of probability is specified by the pdf (probability distribution function). This section is devoted to general properties of tlie pdf in tlie case of discrete and continuous nmdoiii variables. Special pdfs finding e.xtensive application in liazard and risk analysis are considered in Chapter 20. 

Tlie pdf of a discrete randoni variable X is specified by f(x) where f(x) lias the following essential properties  

Consider, for e.xample, a box of 100 transistors containing five defectives. Suppose tliat a transistor selected at random is to be classified as defective or non-defective. Let X denote die outcome, widi X = 0 associated with die drawing of a non-defective and X = 1 associated with die drawing of a defective. Then X is a discrete random variable with pdf specified by 

Tlie pdf of a condnuous random variable X lias die following properties  

Property 1 indicates diat die pdf of a continuous random variable generates probability by integradon of the pdf over die interval whose probability is required. Wlien diis interval contracts to a single value, die integral over the 

The probability that a measurement number will fall with one standard deviation of the true value X is thus 

This probability integral (8.36) can be written using the error function of mathematical physics, denoted 

The error function is also useful for solving chemical diffusion problems (Chapter 13) and thermal conduction problems and the details of the error function and the table of the error fiinction are given in Chapter 13. 

Therefore, there would be a 68% probability that a new measurement number would lie within X (J. 

We present two types of probability distribution discrete and continuous. 

Consider a variable X, which can assume discrete values Xi, X2, ., Xk with respective probabilities (Xi), ( 2). P(Xk). 

Note that X is called a random variable. The function (X) is a discrete probability distribution function. By definition 

Consider a single die. There are six outcomes with the same probability  

Consider two dice. If X is equal to the sum of the face values that the two dice assume in a roll, we can write (Fig. 2.2)  

A large number of probability distributions have been developed in the area of mathematics to perform various types of analysis [15,161. This section presents some of the probability distributions considered useful to perform patient safety-related analysis. 

Although there are a large number of probability/statistical distributions in the published literature, this section presents just five such distributions considered useful for performing oil and gas industry system safety and reliability-related studies [13-15]. 

In Chapter 5 we described a number of ways to examine the relative frequency distribution of a random variable (for example, age). An important step in preparation for subsequent discussions is to extend the idea of relative frequency to probability distributions. A probability distribution is a mathematical expression or graphical representation that defines the probability with which all possible values of a random variable will occur. There are many probability distribution functions for both discrete random variables and continuous random variables. Discrete random variables are random variables for which the possible values have gaps. A random variable that represents a count (for example, number of participants with a particular eye color) is considered discrete because the possible values are 0, 1, 2, 3, etc. A continuous random variable does not have gaps in the possible values. Whether the random variable is discrete or continuous, all probability distribution functions have these characteristics  

I m tired of all this nonsense about beauty being only skin-deep. That s deep enough. What do you want, an adorable pancreas  

So what is probability It is the fraction of all the measurements that will occur between two measurement values. It is defined in this way, rather than to say that it is the fraction of all the measurements that assume a particular value, because of the scaling problem illustrated 

FIGURE 4.2.1 Various probability distributions important in biology. The normal distribution is used for most applications. The t-distribution is used for small sample sizes from a normal distribution. The log normal distribution fits some data better than a normal distribution. The F distribution is used to check equality of variances, and the yj- (chi square) distribution is used to check expected values of data. The curves shown here are for various values of distribution parameters. (From Barnes, J.W., Statistical Analysis for Engineers and Scientists A Computer-Based Approach, McGraw-Hill, New York, 1994.) 

There are independent and conditional (or dependent) probabilities of occurrence. Independent probabilities are ones where there is no linkage among several events, and conditional probabilities are just the opposite. Almost all probabilities in living systems are conditional probabilities. Growth rate, for instance, is usually dependent on the abundance of nutrition. 

There is, however, a range of exhibited growth rates, depending on genetic character and local environment. The probability of the occurrences of different growth rates will depend on many factors, both internal and external to the living system. 

This section lists those statistical functions that require the Statistics and Machine Learning Toolbox in MATLAB to be installed. In pre-2013 versions of MATLAB, this toolbox is called the Statistics Toolbox. 

Detailed information regarding the definitions of the different probability density functions and the meaning of the required variables can be found in Sect. 2.4. Table 7.4 presents a summary of the available functions. 

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Figure 6.10 Probability distributions for two variables input for Monte Carlo...

From the probability distributions for each of the variables on the right hand side, the values of K, p, o can be calculated. Assuming that the variables are independent, they can now be combined using the above rules to calculate K, p, o for ultimate recovery. Assuming the distribution for UR is Log-Normal, the value of UR for any confidence level can be calculated. This whole process can be performed on paper, or quickly written on a spreadsheet. The results are often within 10% of those generated by Monte Carlo simulation. 

If there is insufficient data to describe a continuous probability distribution for a variable (as with the area of a field in an earlier example), we may be able to make a subjective estimate of high, medium and low values. If those are chosen using the p85, p50, pi 5 cumulative probabilities described in Section 6.2.2, then the implication is that the three values are equally likely, and therefore each has a probability of occurrence of 1/3. Note that the low and high values are not the minimum and maximum values. 

Finally, each coefficient were standardized by the division of the sum of all coefficients(2). This definition allows also to regard as the co-occurrence matrix as a function of probability distribution, it can be represented by an image of KxK dimensions. 

Hi) Gaussian statistics. Chandler [39] has discussed a model for fluids in which the probability P(N,v) of observing Y particles within a molecular size volume v is a Gaussian fimction of N. The moments of the probability distribution fimction are related to the n-particle correlation functions and... 

Let a(t) denote a time dependent random process. a(t) is a random process because at time t the value of a(t) is not definitely known but is given instead by a probability distribution fimction W (a, t) where a is the value a (t) can have at time t with probability detennmed by W (a, t). W a, t) is the first of an infinite collection of distribution fimctions describing the process a(t) [7, H]. The first two are defined by... 

In an ensemble of collisions, the impact parameters are distributed randomly on a disc with a probability distribution P(b) that is defined by P(b) db = 2nb db. The cross section da is then defined by... 

Here/(9,(p, i ) is the probability distribution of finding a molecule oriented at (0,cp, li) within an element dQ of solid angle with the molecular orientation defined in tenus of the usual Euler angles (figure B 1.5.10). 

The two exponential tenns are complex conjugates of one another, so that all structure amplitudes must be real and their phases can therefore be only zero or n. (Nearly 40% of all known structures belong to monoclinic space group Pl c. The systematic absences of (OlcO) reflections when A is odd and of (liOl) reflections when / is odd identify this space group and show tiiat it is centrosyimnetric.) Even in the absence of a definitive set of systematic absences it is still possible to infer the (probable) presence of a centre of synnnetry. A J C Wilson [21] first observed that the probability distribution of the magnitudes of the structure amplitudes would be different if the amplitudes were constrained to be real from that if they could be complex. Wilson and co-workers established a procedure by which the frequencies of suitably scaled values of F could be compared with the tlieoretical distributions for centrosymmetric and noncentrosymmetric structures. (Note that Wilson named the statistical distributions centric and acentric. These were not intended to be synonyms for centrosyimnetric and noncentrosynnnetric, but they have come to be used that way.)... 

Wilson A J C 1949 The probability distribution of X-ray intensities Acfa Crystallogr.2 318-21... 

It is instructive to see this in temis of the canonical ensemble probability distribution function for the energy, NVT - Referring to equation B3.3.1 and equation (B3.3.2I. it is relatively easy to see that... 

In either case, first-order or continuous, it is usefiil to consider the probability distribution function for variables averaged over a spatial block of side L this may be the complete simulation box (in which case we... 

This method has been devised as an effective numerical teclmique of computational fluid dynamics. The basic variables are the time-dependent probability distributions f x, f) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a detenninistic local rule. A carefiil choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the fomiation of lamellar phases in amphiphilic systems [92, 93]. 

Monte Carlo simulations generate a large number of confonnations of tire microscopic model under study that confonn to tire probability distribution dictated by macroscopic constrains imposed on tire systems. For example, a Monte Carlo simulation of a melt at a given temperature T produces an ensemble of confonnations in which confonnation with energy E. occurs witli a probability proportional to exp (- Ej / kT). An advantage of tire Monte Carlo metliod is tliat, by judicious choice of tire elementary moves, one can circumvent tire limitations of molecular dynamics techniques and effect rapid equilibration of multiple chain systems [65]. Flowever, Monte Carlo... 

C3.3.5.2 EXTRACTING THE ENERGY TRANSFER PROBABILITY DISTRIBUTION FUNCTION P(E, E)... 

Figure C3.3.11. The energy transfer probability distribution function P(E, E ) (see figure C3.3.2) for two molecules, pyrazine and hexafluorobenzene, excited at 248 nm, arising from collisions with carbon dioxide...

The probability distribution functions shown in figure C3.3.11 are limited to events that leave the bath molecule vibrationally unexcited. Nevertheless, we know that the vibrations of the bath molecule are excited, albeit with low probability in collisions of the type being considered here. Figure C3.3.12 shows how these P(E, E ) distribution... 

Figure C3.3.12. The energy-transfer-probability-distribution function P(E, E ) (see figure C3.3.2 and figure C3.3.11) for two molecules, pyrazine and hexafluorobenzene, excited at 248 nm, arising from collisions with carbon dioxide molecules. Both collisions that leave the carbon dioxide bath molecule in its ground vibrationless state, OO O, and those that excite the 00 1 vibrational state (2349 cm ), have been included in computing this probability. The spikes in the distribution arise from excitation of the carbon dioxide bath 00 1 vibrational mode.

The appropriate quantum mechanical operator fomi of the phase has been the subject of numerous efforts. At present, one can only speak of the best approximate operator, and this also is the subject of debate. A personal historical account by Nieto of various operator definitions for the phase (and of its probability distribution) is in [27] and in companion articles, for example, [130-132] and others, that have appeared in Volume 48 of Physica Scripta T (1993), which is devoted to this subject. (For an introduction to the unitarity requirements placed on a phase operator, one can refer to [133]). In 1927, Dirac proposed a quantum mechanical operator tf), defined in terms of the creation and destruction operators [134], but London [135] showed that this is not Hermitean. (A further source is [136].) Another candidate, e is not unitary. 

Before the limit is taken, the properties of the probability distribution appear to be strange in at least five ways. 

In the q = l limit, the effective temperature equals the standard temperature. Otherwise, adding a constant shift to the potential energy is equivalent to rescaling the temperature at which the canonical probability distribution is computed. 

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

This probability distribution can be found by extremizing the generalization of the entropy Eq. (1) subject to the constraints... 

For a given potential energy function U r ), the corresponding generalized statistical probability distribution which is generated by the Monte Carlo algorithm is proportional to... 

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