Probability, cumulative

For convenience, the probability axis may be split into three equal sectors in order to be able to represent the curve by just three points. Each point represents the average value of reserves within the sector. Again for convenience, the three values correspond to chosen cumulative probabilities (85%, 50%, and 15%), and are denoted by the values ... 

An alternative and commonly used representation of the range of reserves is the proven, proven plus probable, and proven plus probable plus possible definition. The exact cumulative probability which these definitions correspond to on the expectation curve... 

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. 

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 use of the Monte Carlo method in project appraisal was illustrated by Holland et al. [F. A. Holland, F. A. Watson, and J. K. Wilkinson, Chem. Eng., 81, 76-79 (Feb. 4, 1974)]. The cumulative-probability cui-ves of (DCFRR) and (NPV) can never be more accurate than the opinions on which they are based, and comparable accuracy can be obtained by the use of S-shaped cui ves with relatively small computational effort. 

Example 10 Logistics Curve We shall derive the logistics curve representing the ciimiilarive-freqiiency distrihiitions of the normal distrihiition curve defined by Eqs. (9-72) and (9-73). In this case, y varies between a cumulative probability of zero and unity as z varies from — to -l-oo. Since the upper bound is unity, c = 1. From Table 9-10 the area under the right-hand side of the curve between = 0 and z = z may be read. Since the frequency curve is symmetrical about the mean, this is also the area between = 0 and z=z. Hence, the area under the frequency curve, which represents the cumulative probability, is 0.50000 at = 0 and the 80 percentile, for which the area is 0.80000, corresponds to the value = 0.842. We substitute these values into Eqs. (9-92) through (9-94) to give... 

FIG. 9-23 Cumulative probability of a given net present value or less for a project showing normal and Gompertz approximations. 

The cumulative probability distribution of fragments larger than length / is obtained from... 

Commercial software such as MS Excel is useful in this connection being widely available.) Omissions in the ranked values of F, in Table 4.2 reflect the omissions of the data in the original histogram for several classes. As can be judged from Figure 4.10, inclusion of the cumulative probabilities for these classes would not follow the natural pattern of the distribution and are therefore omitted. However, when a very low number of classes exist their inclusion can be justified. 

Parent 2 10 J =f(x) P — J jUJ Cumulative probability Roulette wheel bits... 

Figure 19.8.6. A continuous cumulative probability distribution function.

TABLE 20.5.2 Standard Normal, Cumulative Probability in Rigbt-Hand Tail (for Negative Values of z, Areas Are Found by Symmetry) ... 

Frequency of exceedance per year is obtained by multiplying 0.036, tlie mean annual frequency of release, by the cumulative probabilities in Table 21.5.3. Frequency of exceedance plotted against tlie number of people affected produces a risk curve portraying healtli impact in terms of the frequency with which tlie number of people affected exceeds various amounts. Tlie risk curve is... 

Frequency analysis is an alternative to moment-ratio analysis in selecting a representative function. Probability paper (see Figure 1-59 for an example) is available for each distribution, and the function is presented as a cumulative probability function. If the data sample has the same distribution function as the function used to scale the paper, the data will plot as a straight line. 

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

Figure 13 shows the relationship between the time interval At of passive film breakdown of stainless steel with chloride ions and the logarithms of cumulative probability P(Af) for breakdown at time intervals longer than At. From these results, it is clear that the logarithm of the probability is almost proportional to the time interval, and therefore the cumulative probability for film breakdown follows Poisson s distribution, i.e., the following equation is obtained,... 

Figure 13. Cumulative probability for pit-generation time interval at constant potential of 18Cr-8Ni stainless steel in 0.2 kmol m 3 NaCl solution containing 0.1 kmol nf3 NajSO 23 o, = 0.85 V vs. SCE , =0.95 V vs. SCE. (From N. Sato, J. Etectrochem. Soc. 123, 1197, Fig. 2. Reproduced by permission of The Electrochemical Society, Inc.)...

Figure 6 shows a cumulative probability plot of both the maximum dally and hourly NO2 averages In cities for the 1980-84 time period. The plotted values can be directly compared to the WHO guideline values of 150/tg/m3 for the maximum 24-hour level and 400/tg/m3 for the maximum 1-hour level. In both cases, about 25% of the cities worldwide exceed the guideline values. Based on these proportions of cites with NO2 concentrations above the short-term guideline values. It Is estimated that approximately 15-20 percent of urban residents In North America and Europe are at Increased risk to short-term high NO2 exposures. 

Since one is only rarely interested in the density at a precise point on the z-axis, the cumulative probability (cumulative frequency) tables are more important in effect, the integral from -oo to +z over the probability density function for various z > 0 is tabulated again a few entries are given in Fig. 1.13. 

Figure 1.13. The cumulative probability of the normal distribution. The hatched area corresponds to the difference ACP in the CP plot.

The cumulative probability table can be presented in two forms, namely... 

A table of cumulative probabilities (CP) lists an area of 0.975002 for z -1.96, that is 0.025 (2.5%) of the total area under the curve is found between +1.96 standard deviations and +°°. Because of the symmetry of the normal distribution function, the same applies for negative z-values. Together p = 2 0.025 = 0.05 of the area, read probability of observation, is outside the 95% confidence limits (outside the 95% confidence interval of -1.96 Sx. .. + 1.96 Sx). The answer to the preceding questions is thus... 

Figure 3.8. The transformation of a rectangular into a normal distribution. The rectangle at the lower left shows the probability density (idealized observed frequency of events) for a random generator versus x in the range 0 < jc < 1. The curve at the upper left is the cumulative probability CP versus deviation z function introduced in Section 1.2.1. At right, a normal distribution probability density PD is shown. The dotted line marked with an open square indicates the transformation for a random number smaller or equal to 0.5, the dot-dashed line starting from the filled square is for a random number larger than 0.5.

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