Curve Probability

Expectation curves are alternatively known as probability curves . This text will use the term expectation curve for conciseness. 

Figure 13.19 Cumulative probability curve for an exploration prospect...

Recall a typical cumulative probability curve of reserves for an exploration prospect in which the probability of success (POS) is 30%. The success part of the probability axis can be divided into three equal bands, and the average reserves for each band is calculated to provide a low, medium and high estimate of reserves, //there are hydrocarbons present. 

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

For a more accurate selection of a protective scheme it is essential that the manufacturers provide the probability curves of their shell design for each voltage and rating. 

Figure 26.2 NEMA probability curves (or a case rupture and selection of internal and external fuses...

When the underlying distribution is not known, tools such as histograms, probability curves, piecewise polynomial approximations, and general techniques are available to fit distributions to data. It may be necessary to assume an appropriate distribution in order to obtain the relevant parameters. Any assumptions made should be supported by manufacturer s data or data from the literature on similar items working in similar environments. Experience indicates that some probability distributions are more appropriate in certain situations than others. What follows is a brief overview on their applications in different environments. A more rigorous discussion of the statistics involved is provided in the CPQRA Guidelines. ... 

Wahrscheinlichkeits-gesetz, n. probability law. -kurve, /. probability curve, -rechnung, /. calculus of probabilities, -wert, m. probable value. 

A method of volumetric reserve estimation in which the estimate is expressed as a cumulative probability curve. 

The shape of a frequency distribution curve will depend on how the size increments were chosen. With the common methods for specifying increments, the curve will usually take the general form of a skewed probability curve with a single peak. However, it may also have multiple peaks, as in Fig 2, There are various analytical relationships for representing size distribution. One or the other may give a better fit of data in a particular instance. There are times, however, when analytical convenience may justify one. The log-probability relationship is particularly useful in this respect... 

Figure 1.22. The null and the alternate hypotheses Hq resp. Hi. The normal distribution probability curves show the expected spread of results. Since the alternate distribution ND(/tb, a might be shifted toward higher or lower values, two alternative hypotheses Hi and H are given. Compare with program HYPOTHESIS. Measurement B is clearly larger than A, whereas S is just inside the lower CL(A).

Purpose Display the type I error (a) and the type II error (/3) both as (hatched) areas in the ND(/iREp, Urep) and the ND(/ttest, test) distribution functions and as lines in the corresponding cumulative probability curves. 

Hunter GJ, Silvennoinen HM, Hamberg LM, Koroshetz WJ, Buonanno FS, Schwamm LH, Rordorf GA, Gonzalez RG. Whole-brain CT perfusion measurement of perfused cerebral blood volume in acute ischemic stroke probability curve for regional infarction. 

After analysis of basic statistics and Shapiro-Wilk normality test, it was concluded that the data did not assure the normality conditions necessary to perform certain statistical analysis. To follow usual procedures the data were lognormalised and normality was tested by normal probability curves. 

Many distributions obtained in experimental and observational work are found to have a more or less bell-shaped probability curve. These distributions are described by the normal or gaussian distribution shown in Fig. 2. This theoretical distribution is extremely important in statistics, and its use is not limited to data which are exactly, or very nearly normal. 

If p is an integer between 0 and 100, the pth percentile is the value xp of the random variable X which limits to the right p/lOOths of the surface S under the probability curve (Figure 4.2). 

Alternatively, one may associate significance levels with specific intervals around the mean. For large m, intervals of 1 x s/y/m, 2 x s/yjm, and 3 x s/ /m on each side of x correspond to the 31.7, 4.5, and 0.3 percent significance levels, respectively. They limit 68.3,95.4, and 99.7 percent of the surface enclosed under the density of probability curve. Jargon often refers to these intervals as 1[Pg.197]

The uniformity of the particles as measured by the standard deviation for a normal probability curve was found to be a function of flow rate, nozzle size (better for smaller nozzles), nozzle length (decrease in nozzle length decreases uniformity), etc. A decrease in interfacial tension is insufficient to cause change in uniformity. 

The effects of applying 0.48% of Texspray Compound and 1.09% of Spraycot 8853 are summarized in Table I. In both cases there is a marked drop in the number of particles (an average of 84.8%) and in dust concentration (70.5% for the Texspray and 84.8% for the Spraycot). The dg calculated algebraically decreases from 1.88 un for the control cotton to 1.44 urn for the two lubricated cottons. The cumulative probability curves of the dust emitted from the cottons containing additives become bimodal. These changes in the nature of the cumulative distribution curves suggest that the fraction of particles removed is not constant for all dieuneters. 

When distributions are combined, for example, in joint probability curves, it is important to ensure that the resulting distribution is meaningful, again in terms of what is distributed and with respect to what variable (Suter 1998, p 129). 

The two pulses are presumed separated by the dashed hne given in Figure E6.7.2. For a large number of tanks-in-series (10 or more) the pulse curve is close to a Gaussian probability curve. From equation (6.12), there are... 

In the above equation, 0.40 In(.r) represents the slope of the probability curve. Reducing it to 0.30 In(.r) would translate to a 23% reduction in VOCs released to the environment. 

Normally one might expect that if the transition probability vanishes on resonance it also vanishes off resonance. However, such is not the case. When the transition probability is calculated off resonance, by numerically solving Eqs. (14.16) using a Taylor expansion method, it is nonzero for both v E and v 1E.14,16 In Fig. 14.6 we show the transition probabilities obtained using two different approximations for v E, and vlE for the 17s (0,0) collisional resonance.16 To allow direct comparison to the analytic form of Eq. (14.21) we show the transition probabilities calculated with EAA = VBB = 0. For these calculations the parameters ju2l = pLz, = 156.4 ea0, b = 104ao, and v = 1.6 x 10-4 au have been used. The resulting transition probability curves are shown by the broken lines of Fig. 14.6. As shown by Fig. 14.6 these curves are symmetric about the resonance position. The vlE curve of Fig. 14.6(b) has an approximately Lorentzian form, but the v E curve of Fig. 14.6(a), while it vanishes on resonance as predicted by Eq. (14.24), has an unusual double peaked structure. 

A break was observed in the translational energy probability curve for CS radicals at the thermochemical threshold for the production of S( D). It was then assumed that only atoms were produced below this threshold, while only 3p atoms were produced above this threshold. With this assumption they were able to derive a lower limit of 0.25 for the S(3p)/S(lD) ratio. They were also able to calculate a vibrational distribution from the TOF curve that agrees with the LIF measurements. The overall conclusions of this paper are in direct disagreement with the work of Addison et al. 

Fig. 11. Demonstrating the probability curve shift due to the Ra/Rb ratio change.

In summary, my view is that the fundamental cause for superplasticity is electronic in origin which has to do with the probability curves for the formation of compounds. This in turn creates the instability of the compounds and results in the ultra small grain size. Then, on the application of tensile stress, the plastic deformation is purely mechanical and has nothing to do with electrons. This is completely different from that observed in the normal plasticity as described above. The cause and mechanism for super-plasticity and normal plasticity are therefore fundamentally different. The phenomenon of superplasticity therefore can be viewed stepwise as follows ... 

The transition from the ground to the excited state, where the excitation goes from v = 0 (in the ground state) to v = 2 (in the excited state), is the most probable for vertical transitions because it falls on the highest point in the vibrational probability curve in the excited state. Yet many additional transitions occur, so that the fine structure of the vibronic broad band is a result of the probabilities for the different transitions between the vibronic levels. 

The probability curve is the same as in Figure 2.3, but the gray region in this case is situated on the opposite side of the action level, and the false acceptance and false rejection decision errors have changed places on the probability curve. With low probabilities of decision error, soil with sample mean concentrations significantly below 100 mg/kg will be accepted as backfill, soil with sample mean concentrations clearly exceeding 100 mg/kg will be rejected. However, if the true mean concentration is slightly above the action level, for example, 110 mg/kg, and the sample mean concentration is less than 100 mg/kg (e.g. 90 mg/kg), then the project team will be likely to make a false acceptance decision error. A consequence of false acceptance... 

The planning team is now prepared to assign decision error limits to false acceptance and false rejection decision errors. A decision error limit is the probability that an error may occur when making a decision based on sample data. The probability curve tells us that the highest probability of error exists in the gray region this error goes down as the mean concentrations move away from either side of the action level. The probability curve reflects our level of tolerance to uncertainty associated with a decision or, conversely, level of confidence with which a decision will be made. 

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