Error probability function

A typical distribution of errors. The bar graph represents the actual error frequency distribution 73(e) for 376 measurements the estimated normal error probability function P(e) is given by the dashed curve. Estimated values of the standard deviation cr and the 95 percent confidence limit A are indicated in relation to the normal error curve. 

A probability function derived in this way is approximate the true probability function cannot be inferred from any finite number of measurements. However, it can often be assumed that the probability function is represented by a Gaussian distribution called the normal error probability function,... 

The dashed curve in Fig. 3 represents a normal error probability function, with a value of (T calculated with Eq, (13) from the 376 errors... 

The usual assumptions leading to the normal error probability function are those required for the validity of the central limit theorem The assumptions leading to this theorem are sufficient but not always altogether necessary the normal error probability function may arise at least in part from circumstances different from those associated with the theorem. The factors that in fact determine the distribution are seldom known in detail. Thus it is common practice to assume that the normal error probability function is applicable even in the absence of valid a priori reasons. For example, the normal error probability function appears to describe the 376 measurements of Fig. 3 quite well. However, a much larger number of measurements might make it apparent that the true probability function is slightly skewed or flat topped or double peaked (bimodal), etc. 

Thus, the function P(m) is called the error probability function and represents the distribution of the probability over the measured values. Such a distribution is called a probability aggregate. Every measured value is an element of that aggregate. 

Figure 12-2 Distribution of measured values showing the error probability function P(m).

The distribution of the /-statistic (x — /ji)s is symmetrical about zero and is a function of the degrees of freedom. Limits assigned to the distance on either side of /x are called confidence limits. The percentage probability that /x lies within this interval is called the confidence level. The level of significance or error probability (100 — confidence level or 100 — a) is the percent probability that /X will lie outside the confidence interval, and represents the chances of being incorrect in stating that /X lies within the confidence interval. Values of t are in Table 2.27 for any desired degrees of freedom and various confidence levels. 

Random Measurement Error Third, the measurements contain significant random errors. These errors may be due to samphng technique, instrument calibrations, and/or analysis methods. The error-probability-distribution functions are masked by fluctuations in the plant and cost of the measurements. Consequently, it is difficult to know whether, during reconciliation, 5 percent, 10 percent, or even 20 percent adjustments are acceptable to close the constraints. 

Human Error Probability The probability that an error will occur during the performance of a particular job or task within a defined time period. Alternative definition The probability that the human operator will fail to provide the required system function within the required time. 

Great simplification is achieved by introducing the hypothesis of independent reaction times (IRT) that the pairwise reaction times evolve independendy of any other reactions. While the fundamental justification of IRT may not be immediately obvious, one notices its similarity with the molecular pair model of homogeneous diffusion-mediated reactions (Noyes, 1961 Green, 1984). The usefulness of the IRT model depends on the availability of a suitable reaction probability function W(r, a t). For a pair of neutral particles undergoing fully diffusion-con-trolled reactions, Wis given by (a/r) erfc[(r - a)/2(D t)1/2] where If is the mutual diffusion coefficient and erfc is the complement of the error function. 

The error in D is a strong function of A[H+]/[H+], A[HA]/[HA] and A hpl/ HpL > small changes in [HA],9hpl or pH can produce large deviations of the D value. The principal source of error probably arises from the pH measurement. For example the error in the normalized D and D values for a trivalent two-order complex may be expressed ... 

The ALLOC method with Kernel probability functions has a feature selection procedure based on prediction rates. This selection method has been used for miik >s5) and wine > data, and it has been compared with feature selection by SELECT and SLDA. Coomans et al. suggested the use of the loss matrix for a better evaluation of the relative importance of prediction errors. 

It is the root-mean-square error expected with this probability function ... 

So far the discussion has dealt with the errors themselves, as if we knew their magnitudes. In actual circumstances we cannot know the errors by which the measurements Xj deviate from the true value Xq, but only the deviations (x,- — x) from the mean T of a given set of measurements. If the random errors follow a Gaussian distribution and the systematic errors are negligible, the best estimate of the true value Aq of an experimentally measured quantity is the arithmetic mean x. If you as an experimenter were able to make a very large (theoretically infinite) number of measurements, you could determine the true mean /jl exactly, and the spread of the data points about this mean would indicate the precision of the observation. Indeed, the probability function for the deviations would be... 

The normal probability function as expressed by Eq. (14) is useful in theoretical treatments of random errors. For example, the normal probability distribution function is used to establish the probability Hthat an error is less than a certain magnitude 8, or conversely to establish the limiting width of the range, —8 to 8, within which the integrated probability P, given by... 

A design that meets the information axiom is called a robust design (see Technique 38) because it maximizes the probability that it will meet its specihcations (process variables, or PVs) on an ongoing basis. For example, a process variable like tensile strength or data-entry errors can function as intended along a spectrum from zero to absolute perfection but in reality it always functions somewhere in between the two extremes. 

Setting the error density function proportional to the exponential function just given, and adjusting to unit total probability, one obtains... 

Fortunately, the necessary mathematics for higher numbers of stages are aU worked out for us in the mathematics of probability. MTien the number of stages is large, such a curve becomes the normal curve of error. By substituting Kj K -f- 1) for and 1/(1 -f- K) for q in the probability function, ... 

Error probabilities. Generally, a class of functions regarded as very small can be given, and one requires that the function 1 - P C ys jars, i group) is very small for all attackers A e Attacker jlass(Scheme, Req). 

I also expect that computational restrictions only make sense in combination with allowing error probabilities, at least in models where the complexity of an interactive entity is regarded as a function of its initial state alone or where honest users are modeled as computationally restricted. Then the correct part of the system is polynomial-time in its initial state and therefore only reacts on parts of polynomial length of the input from an unrestricted attacker. Hence with mere guessing, a computationally restricted attacker has a very small chance of doing exactly what a certain unrestricted attacker would do, as far as it is seen by the correct entities. Hence if a requirement is not fulfilled information-theoretically without error probability, such a restricted attacker has a small probability of success, too. 

Note that the error probability in Part c) is only taken over key generation (in the functional version), and not over the messages. Hence, as anticipated in Section 7.1.3, arbitrary active attacks have been hidden in the quantifier over the message sequence If the keys are not in Badg N), there is no message sequence for which authentication is not effective, hence it does not matter whether the attacker or an honest user chooses the messages, and whether adaptively or not. 

This notion of exponentially small is defined by an explicit upper bound, as in previous conventional definitions. A benefit is that one can decide what error probability one is willing to tolerate and set a accordingly. A disadvantage is that the sum of two exponentially small functions need not be exponentially small in this sense, i.e., this formalization does not yield modus ponens automatically (see Section 5.4.4, General Theorems ). Hence different properties have to be defined with different error probabilities, and some explicit computations with O are needed. 

Two security parameters have been used here. Usually, there is only one (or even none — then the security is measured as a function of the length of the input K exclusively), and strictly exponential decrease of the error probability in the soundness is not required. In the present application, however, it is needed. Anyway, most existing zero-knowledge proof schemes are repetitions of one basic round, and the error probability decreases exponentially with the number of rounds. Hence the number of such rounds would be a linear function of a. 

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