Application of the Negative Binomial/Pascal Distribution in Probability Theory to Electrochemical Processes

The subject of interest here is a set of random observations falling into only two categories: success and failure, their single-event probability being constant and independent of any previous occurrence during a sequence of such events. The probability distribution of their number, known generally as Bernouilli trials, is called the negative binomial distribution (NBD) or the Pascal distribution. In this context, success and failure are completely relative concepts, their physical meaning defined by the experimenter or analyst: the appearance of a substandard product, for instance, among acceptable ones may well be considered success by a quality controller whose objective is the identification of substandards. This distribution serves in general for finding the probability of an exact number, or at least, or at most a certain number of failures observed upon so many successes (or vice versa, depending on the chosen definition of success and failure). Its special form, related to the appearance of the first success (or failure) is called the geometric distribution.


Important parameters of the NBD
Table 2 contains parameters related to the first four statistical moments, with original notations adjusted to comply with the notation scheme in Doherty (1990)

Estimation of the single-event success probability from experimental observations
If p is not known a-priori, Evans et al. (2000) recommend the unbiased-method estimator p*(UB) = (k 0 -1)/(n 0 + k 0 -1) (9) and the maximum likelihood-method estimator p*(ML) = k 0 /(n 0 + k 0 ) (10) based on available experimental data.The zero subscript refers to anteriority, and the asterisk indicates that Eqs.( 9) and ( 10) are estimators of the unknown ("true") population parameters.When n 0 and k 0 are small, discrepancy between the two estimates can be very large.Conversely, if k 0 >> 1, the two estimates are essentially equal.

The electrolytic reduction of acrylonitrile (ACN) to adiponitrile (ADN)
In one of the major electroorganic technologies the 82-90% selectivity of ADN production was ascribed primarily to efficient control of major byproducts: propionitrile and a C 9 H 11 N 3trimer, as well as to the non-electrochemical formation of biscyanoethylether (Danly & Campbell, 1982).It is assumed here that in a hypothetical pilot -plant scale operation of a modified ADN process, batches produced within a set time period will be tested for ADN selectivity S immediately upon production.Denoting the random number of batches exhibiting S > 85% as N, and the single-event probability of S ≤ 85% as p, the probability that exactly N = n number of batches will exhibit S > 85% while k number of batches will exhibit the opposite in a sequential sampling is given by Eq.( 1).Similarly, the probability that not more than N = n' batches will exhibit S > 85% is given by Eq.( 3).The finding of a "substandard" batch is considered to be success in this context, since the elimination of such batches would be the ultimate goal.

A nickel-iron alloy plating process
A novel NiFe alloy plating process is envisaged to have been carried out in several sequential experiments adhering to a tightly controlled Ni 2+ /Fe 2+ ionic ratio in the cell electrolyte kept within a narrow experimental temperature range.Defining alloy deposits of poor quality as a success (by the same reasoning as in Section 3.1), the experiments are assumed to indicate eight failures prior to three successes.In the absence of any knowledge of single-event success probabilities, the latter can be estimated to be p* (UB) = (3-1)/(8+3-1) = 0.2 [Eq.( 9)] and p* (ML) = 3/(8+3) = 0.2727 [Eq.( 10)].

An electrolytic nanotechnological-size cadmium plating process with tagged Cd 2+ ions
In a hypothetical study of its mechanism, a cadmium plating process is assumed to proceed until a monolayer of about 100 discharged Cd 2+ ions (ionic radius = 0.097 nm; Dean, 1985) has fully been formed on an approximately 3 (nm) 2 deposition area by a 300 nA current pulse of 10 ms duration (corresponding to a current density of about 100 mA/cm 2 in conventional plating technology).Ten percent of the ions in the electrolyte are tagged (e.g., radioactively) for monitoring purposes.Their arrival to the surface with respect to untagged ions may be considered a sequence of Bernouilli trials with probability mass function and cumulative mass function in view of the fundamental relationship between the gamma function and factorials: Γ ( x + 1) = x!Selected individual and cumulative probabilities pertaining to the arrival of the first five tagged ions to the electrode surface are shown in deposition area.N is the random variable denoting the number of untagged ions that have arrived at the surface along with the arrival of the K = 5 tagged ions.

An aluminum anodizing process
In a typical conventional anodizing process (Pletcher & Walsh, 1990) using a 50 g/dm 3 chromium acid electrolyte, high corrosion-resistance opaque grayish white (possibly enamel) films are produced at an electrolyte temperature of 50 0 C. To maintain a deposition rate of 13 -20 μm/h the cell potential is gradually raised from zero to 30 V while the current density reaches an asymptote within the 10 -15 mA/cm 2 range.
In a hypothetical research project with the objective of maintaining higher deposition rates, modified chromic acid concentration, and lower temperatures (i.e.lower energy input), a tolerance domain consisting of a combined range of acceptable operating variables in each bath (cell) is to be established, in order to meet new acceptable performance criteria.
Considering bath -to -bath performance levels as Bernouilli trials, q is defined as the fraction of acceptable and p = 1 -q as the fraction of unacceptable quality (i.e."success" in NBD parlance).Consequently, if the random variable N is the number of acceptable baths obtained prior to finding K unacceptable baths, the probability of finding exactly N = n acceptable baths on the k-th unacceptable bath is given by Eq.(1) rewritten as and specifically, when k = 1, Selected probability values are displayed in Table 6.6. Failure/success probabilities and principal distribution parameters in Section 3.4, related to the arrival of anodizing baths of unacceptable quality.N is the number of acceptable baths found at the k-th arrival.

Non-conducting oxide layer formation on Ti-MnO 2 and Ti-RuO 2 anodes
The possibility of non-conducting oxide layers forming on untreated titanium -manganese oxide and titanium -ruthenium oxide substrate has been known to be a source of failure for dimensionally stable anodes (DSA) over several decades ( Smyth, 1966).In this illustration a recently developed experimental DSA is supposed to carry a prohibiting additive embedded in the conducting oxide matrix in order to reduce the presence of nonconductors.The reliability of the additive is estimated to be 98%, i.e. 2% of the DSA are believed to be susceptible to failure.In two independent tests, complying with Bernouilli trial conditions, twelve anode samples were taken randomly for each test from a large ensemble of anodes, finding four (Test 1) and five (Test 2) defective specimens.The defective ratios 4/12 = 0.333 and 5/12 = 0.417 might imply a prima facie rejection of the 2%-defectives-at-most claim by the anode manufacturer, as a questionable ("primitive") means of judgment.A careful NBD -based approach employing cumulative distributions (to account for all possible, not just the actually observed outcome) yields probabilities (via Eq.( 2) and Eq.( 4 which demonstrate, at a negligible numerical difference, an extremely low likelihood of finding up to eight and up to seven anode specimens, respectively, due to random effects.Hence, the 2% claim appears to be highly questionable.

Computation of intermediate probabilities
The probability of a negative binomial variable N occurring (due to random causes) from a value N = n 1 to value N = n 2 can be expressed in the simplest form as Under specific conditions, tabulations of the incomplete beta function (e.g.Beyer, 1968) may be employed for a quick estimation of probabilities (Appendix C).

Analysis of the contents of Sections 3.1 -3.5
The primary role of Tables 2 -6 resides in reaching a decision whether or not experimental observations indicate the presence of non -random effects causing the observed results.For the ADN -process in Section 3.1, the entries in Table 3 indicate that if, for instance, ten batches with S > 85% selectivity were found along with 1 -5 batches with S ≤ 85%, this finding would suggest a rather strong promise for the new process, inasmuch as there would be at most an about 0.5% chance for such a result arising from random reasons (at least in the 1 ≤ k ≤ 5; 0.2 ≤ p ≤ 0.8 ranges).Conversely, if e.g. both selectivity ranges were equally probable ( p = q = 0.5), the finding of exactly two batches with S > 85% selectivity would be at most slightly promising for the new process, since an about 12 -19% probability exists for random causes.However, if there were, for instance, at least five S > 85% observations, the process would be judged highly promising.
In a somewhat different manner, the modified process could also be deemed to be acceptable, if the coefficient of variation CV related to the GD describing the appearance of the first S ≤ 85% batch, one of the absolute measures of dispersion, and defined as the ratio of the standard deviation to the mean of the distribution, differed from unity only within a pre-specified (small) fraction.From Table 2, CV = 1/√q when k = 1, and it follows that as p becomes progressively smaller, both q and CV approximate unity, the latter from values above.Stipulation of a not more than 10% downward difference from unity (i.e.CV ≤ 1.1) would specify p ≈ 0.2, as the highest acceptable single -event probability of S ≤ 85% selectivity; put otherwise, the appearance of the first S ≤ 85% batch after four S > 85% batches would imply promise for the new process.If the stipulation were a more stringent 1% downward difference from unity, CV ≤ 0.1 would prescribe p ≈ 0.02, i.e. an only 2% single event probability of S ≤ 85% selectivity.
As shown in Section 3.2, when single-event probabilities are not available from prior sources, their numerical values are highly sensitive to the method of parameter estimation.The largest individual probability: p * ML = 0.0867 occurring at mode = 5 signals at most an approximately 9% maximum chance for randomness-related observation of good quality deposits along with the observation of three low -quality deposits.It follows that the novel process can be considered effective.Cumulative probabilities yield the same qualitative result.If p * UB = 0.2 is accepted for the single-event success (i.e.poor deposit quality), experimental observations of failures (i.e.good deposit quality) with three successes would suggest process reliability in face of an about 6% chance of random occurrence.By contrast, randomness-related probability of up to ten failures along with three successes being almost 50%, the effectiveness of the new process would appear to be rather dubious, if such www.intechopen.com Application of the Negative Binomial/Pascal Distribution in Probability Theory to Electrochemical Processes 39 occurrences were observed experimentally.If p * ML ≈ 0.27 is accepted, the almost two-thirds probability of finding up to ten failures due to random effects essentially rules out the new process as a viable alternative to the existing one.Since these estimators come from a single experiment, the conclusions should be accepted only provisionally until further experiments will have established a wider data base.
It is instructive to consider situations with a relatively high number of experimentally observed successes.Assuming, for the sake of argument, n 0 = 3 and k 0 = 10, the single event success probabilities p * (UB) = 9/12 = 0.7500 and p * (ML) = 10/13 = 0.7692 are expected to deliver near -identical probability estimates.As shown in 0.9 6 4 0.9 6 4 Table 7. Selected probabilities based on single event probability estimators in Section 3.2 .In columns 5 -7 the first entry is UB -based, the second entry is ML -based.In column 7 "9 6 " denotes six consecutive nines to indicate the extent of closeness to unity (i.e., certainty).
In the ion-tagging scenario of Section 3.3, and as shown in Table 5, the largest individual probability of untagged ion arrival, predicted by the mode (n = 36) is only about 2%, but there is a nearly 40% cumulative chance that up to 36 untagged ions are in fact at the surface on that particular occasion.At the mean value n = 45, the cumulative probability is about 57%.Given skewness (1 + 0.9)/√(0.9)(5)≈ 0.90, and kurtosis 3 = 6/5 + (0.1) 2 /[(0.9)(5)]≈ 4.2, the asymmetric distribution may be considered to be moderately skew (Bulmer, 1979a) and somewhat leptokurtic (Bulmer, 1979b), with respect to the skew-free normal (Gaussian) distribution, serving as reference, whose kurtosis is exactly 3.
Monitoring the arrival of untagged ions in Section 3.3 prior to the presence of the first tagged ion at preset values of their probability may also be an important objective of process analysis.Values obtained via Eq.( 5) and Eq.( 6) have been rounded to the nearest upward or downward integer in Table 8.The entries in the second column are given by the numerical form of Eq.( 5) : n = -21.8543-9.949122 ln(P), and in the third column by the numerical form of Eq.( 6) : n' = -9.49122ln(P).No untagged ions can be expected to arrive prior to the first tagged ion at a probability higher than about 9%, whereas cumulative probabilities of their prior arrival increase with their number.While there is only a 20% probability that one (or no) untagged ion precedes the first tagged ion, it is almost certain for 65 untagged ions to do www.intechopen.comso.At a higher single -event probability of tagged ion arrival, the number of previously arrived untagged ions is, of course, smaller: when, e.g.p = 0.2, about twenty or thirty such ions may be expected to arrive at a 99% and 99.9% probability, respectively, in contrast with forty three and sixty five, when p = 0.1 .Table 8.Selected numbers of untagged ions that have arrived at the electrode surface in Section 3.3 prior to the arrival of the first tagged ion at (arbitrary) preset probabilities (p = 0.1; q = 0.9). 1 Eq.( 5); 2 Eq.( 6) The effect of the single-event success probability is also illustrated in Table 9 for the aluminizing baths of Section 3.4 .If this probability is low (p = 0.2) for unacceptable baths, the cumulative probabilities show that a relatively large number of acceptable baths can be produced prior to the first unacceptable bath.Conversely, only a relatively small number of acceptable baths can be expected, if this probability is high, before the arrival of the first unacceptable bath.Individual probabilities are much smaller: if, e.g.p = 0.1, it is essentially certain that (at least) up to sixty five acceptable baths would be found before the first unacceptable bath, but the chances of finding exactly sixty five baths is only (0.1) (0.9 65 ) ≈ 10 -4 .Single-event probabilities pertaining to the event of acceptable (or unacceptable) baths can be readily computed by solving the nonlinear algebraic equation q n -q n+1 = P[q]; n fixed (18) The number of observed acceptable baths determines the largest probability obtained from the dP/dq = 0 condition, resulting in and is illustrated in Table 10.Vacancy in the blocks associated with n = 4, 5 and 6 indicates that the P[q] = 0.10 requirement cannot be satisfied inasmuch as the P max values: 0.0819; 0.0670 and 0.0567 are significantly below 0.10.As shown in the last row of Table 10, the P[q] = 0.1 stipulation can be barely met even at n = 3, but it can be comfortably satisfied at n = 2, (and at n = 1,not shown explicitly; Eq.( 18) readily yields q = 1/2 + √0.15 = 0.8873, hence p = 0.1187.Also, from Eq.( 19), P max = 1/4 = 0.25, hence q max = p max = 0.5).

P[q] n
Single  10.Single event success probabilities at selected values of P[q] and n in Eq.( 18), and the largest attainable P[q] = P max via Eq.( 19) in Section 3.4 A more sophisticated (and time consuming) determination of single-event probabilities from a set of experimental observations would require (nonlinear) regression techniques applied to Eq.( 18) carrying measurement replicates of the number of acceptable baths that precede the first unacceptable bath.
Single event probability q of finding an anode of unacceptable quality Table 11.The cumulative probability that, in Section 3.5, finding up to eight (Test 1, K = 4), and up to seven (Test 2, K = 5) anodes of acceptable quality is due to random effects.
Using Table 11 as a guide, a process analyst setting a cumulative probability of about two percent as the acceptance threshold for the 2% -defectives claim, would be inclined to accept it when (unknown to the analyst) q ≥ 0.93 in Test 1, and when (again unknown to the analyst) q ≥ 0.88 in Test 2. A different (and more exacting) analyst setting the claimacceptance threshold to about 0.05% would accept the claim when (again unknown to the analyst) q ≥ 0.965.
In the alternative fixed sample plan FSP approach (Blank, 1980) the size of the anode ensemble is fixed prior to testing.While it is, in principle, a matter of arbitrary choice, it should not be smaller than the mean of the failure occurrences plus the number of successes (i.e. the mean number of acceptable anodes plus the number of defective anodes).
Another decision scheme might be based on the probability that, in face of the 2%defectives claim the number of acceptable anodes is between two pre-specified values.If, for the sake of argument, these values are 22 and 25 inclusive, and assuming that when a sufficiently large number of anodes were tested three defectives were found, the probability calculated via Eq.( 17): would permit inference of effectiveness for the new process.Similarly, if the decision criterion is based on the incidence of the first defective anode upon the appearance of N anodes of acceptable quality, and it is stipulated that this event occur from the 21 st to the 24 th test inclusively, the probability P[21 ≤ N ≤ 24; K = 1] = (0.02)(0.98 21 + 0.98 22 + 0.98 23 + 0.98 24 ) = 0.98 21 -0.98 25 = 0.0508 ( 22) would lead to the same conclusion.If the somewhat more stringent condition for the appearance of the first defective anode between the 14 th and the 17 th test inclusive were set, Eq.( 22) would be rewritten as P[14 ≤ N ≤ 17; K = 1] = (0.02)(0.98 14 + 0.98 15 + 0.98 16 + 0.98 17 ) = 0.98 14 -0.98 18 = 0.0585 ( 23) and a process analyst, inclined to accept the 2 % -d e f e c t i v e s c l a i m o n l y u p t o a 4 % cumulative probability due to random effects would most likely question the effectiveness of the process.The level of acceptance chosen by the process analyst is an admittedly subjective element in the decision process, but not even quantitative sciences can be fully objective at all times.This statement is all the more valid for applied probability methods as probabilities are prone to be influenced by individual experiences.

Caveats related to negative binomial distributions
One inviting pitfall in dealing with NBD -related probability calculations would be the attempt to apply combination strings where they do not apply.A case in point can be the 2% -defectives claim in Section 3.5 if, in computing e.g., the probability of finding eight acceptable anodes in the presence of four defective ones, the erroneous path: C(u;8)C(v;4)/C(u + v;12) were chosen.The latter provides the probability of selecting eight items out of u identical items simultaneously with selecting four items out of v identical items, the u and v items being of a different kind, with replacement.Apart from the conceptual error in this scheme, the arbitrary choice of u and v predicts widely different numerical values( if, e.g.u = 60 and v = 20, C(60;8) C(20;4)/C(120; 12) = 0.00117 is considerably different from, e.g., C(45;8) C(6;4)/C(51;12) = 0.0204).
Equally important is the stipulation of mutual independence of the Bernouilli trials.In the context of Section 3.5, e.g., the anode -producing process should have no "memory" of the quality of any previously produced specimen.Otherwise, the NBD -based probability calculations would produce biased, i.e. statistically unreliable results.Similar considerations apply as well to the other illustrative examples.Event interdependence would necessitate working with conditional probabilities; if E 1 were the event of a success and E 2 the event of failure, P(E 1 /E 2 ) would be the probability of a success occurring when a failure has occurred.This topic is further explored in the sequel.
6. Interdependence effects: the role of conditional probabilities and Bayes' theorem

Fundamental concepts
In terms of conventional set theory, the intersection (A∩B) denotes the simultaneous existence ("coexistence") of two events, and (A/B) the existence of event A on the condition that event B has happened (is known to have happened).The probability expression P(A∩B) = P(A/B)P(B) = P(B/A)P(A) serves as the underpinning of Bayes' theorem (also known as Bayes' rule) involving conditional probabilities, written in a general form as P(Ak/B) = P(B/Ak)P(Ak)/[P(B/A1)P(A1) + P(B/A2)P(A2) +…+P(B/Ak)P(Ak) +…+P(B/An)P(An)] ( 24) provided that the A k , k = 1,…,n events are exclusive (i.e., independent of one another) and exhaustive (i.e., at least one of the events occurs).In the simpler instance of dual event sets (A, and A': not A), (B, and B′: not B) it follows from Eq.( 24) that The denominator in Eq.( 24) and Eq.( 25) yields the overall probability of event B occurring.Independence of events A k and B may consequently be defined as P(B/A k ) = P(B) and P(B/A) = P(B), etc.Similarly, the probability of intersecting events is simply the product of their individual probabilities: P(A∩B) = P(A)P(B), or equivalently, P(B∩A) = P(B)P(A), in the case of independence.
Employing similar arguments, the rest of the posterior probabilities may be written as with the understanding that P(A/B) + P(A'/B) =1; P(A/B') + P(A'/B') = 1.

Application to the aluminum anodizing process in Section 3.4
In order to illustrate the Bayes' theorem -based approach, the assumption is made that the single event success probability p, i.e., the probability of an anodizing bath performing in an unacceptable manner, does not have a strictly defined value; in fact, it varies with a-priori experienced conditional probabilities.In this context A is defined as the event of p being lower or equal to a "believed" or preset value p * , and the complementary event A' is that p > p * .B is the event of the range of acceptable operating variables falling within a specified tolerance domain, and B' is the event of the range being outside the domain.For the sake of www.intechopen.com Application of the Negative Binomial/Pascal Distribution in Probability Theory to Electrochemical Processes 45 numerical demonstration, the prior probabilities P(A) = 0.85; P(B/A) = 0.97 and P(B/A') = 0.04 are set.They state, on the basis of some prior experience, respectively, that (i) there is an 85% chance for p ≤ p * ; (ii) a 97% chance that if p ≤ p * , the range of operating variables ROV falls within the specified tolerance domain STD, and (iii) a 4% chance that if p ≤ p * , the range is outside the domain.The calculations assembled in Table 12 reinforce the p ≤ p * hypothesis inasmuch as P(A/B) is near 100%, whereas P(A'/B) is less than 1%.The probabilities P(A/B') ≈ 0.15 and P(A'/B') ≈ 0.85 strongly imply that if the ROV were outside the STD, the p ≤ p * hypothesis would be essentially untenable.Table 12.Application of Bayes' theorem to the anodizing bath in Section 3.4 It is instructive to examine the sensitivity of these results to variations in P(A), P(B/A) and P(B/A') while other pertinent probabilities remain constant.The relationships lead to a wealth of useful inferences.An increase in prior probability P(A) produces a general tendency toward unity for posterior probabilities P(A/B) and P(A/B'), although the effect on the latter is less rapid.If P(A) is close to unity, Bayes' theorem predicts little difference with respect to event B or event B'.
In a somewhat more complicated situation several ranges of single-event probabilities would exist with related conditional probabilities.This case is illustrated by assuming three adjacent success probability ranges: 0.02 ≤ p < 0.05; 0.05 ≤ p < 0.1; 0.1 ≤ p < 0.2, their events denoted as A 1 ; A 2 ; A 3 ; ,respectively.With prior probabilities P(A 1 ) = 0.4; P(A 2 ) = 0.5; P(A 3 ) = 0.07, they represent a firm "belief" in p falling between 0.02 and 0.1, but considerable uncertainty about where it can be expected between these bounds.Similarly, the conditional probabilities, assumed to be P(B/A 1 ) = 0.3; P(B/A 2 ) = 0.4; P(B/A 3 ) = 0.15, reflect this state of affairs.Then, in Eq.( 24) the probability of the OVR falling within STD, namely P(B) = 0.3x0.4+ 0.4x0.5 + 0.15x0.07= 0.3305 raises serious doubt about the properness of OVR position, and not surprisingly, the only firm conclusion that can be drawn from the posterior probabilities: P(A 1 /B) = 0.3x0.4/0.3305= 0.3631; P(A 2 /B) = 0.4x0.5/0.3305= 0.6051 and P(A 3 /B) = 0.15x0.07/0.3305= 0.03118 is the rather weak likelihood of p falling between 0.1 and 0.2 due to random effects.Let the anodizing process produce ten baths of acceptable, and three baths of unacceptable quality in a subsequent test.Table 13 indicates that inside the domain of single-event success probabilities p must be lower than 0.07 in order to judge the anodizing process acceptable if P[N = 10;K = 3] ≤ 0.05 is stipulated as a reasonable acceptance criterion for the process.Under such circumstances, acceptance would at worst carry with it an about 13% possibility of random effects accompanying the process analyst's decision.

Normal distribution -based approximations to NBD with large success occurrences
Rephrasing Theorem 4 -13 (Arnold, 1990b) concerning large success numbers K, the random variable (N k -μ)/σ closely approximates the standard random normal variate Z as K increases, and in the limit, with mean μ ≡ k(1 -p)/p and variance σ 2 ≡ k(1 -p)/p 2 .Taking into account the conventional continuity correction required when a discrete distribution is approximated by a continuous distribution, it follows that where and Φ(z) is the cumulative standard normal distribution function, tabulated extensively in the textbook literature and monographs on probability and statistics.
Table 14 demonstrates that even at a relatively small number of successes the normal approximation to NBD is well within the same order of magnitude, albeit not uniformly so; this is also seen in the instance of cumulative probabilities.As a case in point, from Table 14: P[0 ≤ N ≤ 7] = 0.2946 is obtained employing rigorous NBD theory, whereas the normal approximation via the sum (0.0076 + 0.0017 + 0.0178 + 0.0270 + 0.0358 + 0.0469 + 0.0615 + 0.0700) = 0.2783 differs from the rigorous value only by a relative error of about -6%.The simpler "shortcut": P[0 ≤ N ≤ 7] ≈ Φ [(7.5 -10)/√20] = Φ (-0.56) = 0.2877 equally qualifies as a good approximant on account of a relative error of about -2.3%.The poorer approximation via Φ [(7 -10)/√20] ≈ Φ (-0.67) = 0.2514 with a -14.7% relative error is the price to pay if the continuity correction is neglected.The last column in Table 14 demonstrates the unevenness of the error magnitudes.Using the relative error, or the error magnitude as a measure of approximation quality is the analyst's decision.

Conclusions
The illustrative examples, albeit not exhaustive, demonstrate the potential of NBD theory for analyzing a wide variety of electrochemical scenarios from a probabilistic standpoint.In view of a still rather limited employment of probability -based and statistical methods in the electrochemical research literature, a major intent of the material presented here is a "whetting of appetite" by stimulating cross fertilization between two important disciplines.
There remains much more work to be done in this respect.

Acknowledgment
The manuscript was written on equipment provided by the Natural Sciences and Engineering Research Council of Canada (NSERC), and the Department of Chemical Engineering of the University of Waterloo.

Appendix A
The Bernouilli trial is a statistical experiment involving a binomial event with success probability p and failure probability q = 1 -p (success and failure are opposite but arbitrarily defined events).Each trial is independent of any previous trial and p and q do not change from trial to trial.The exponent x in the pmf can take only two values, namely x = 0 (failure with probability q), or x = 1 (success with probability p).This is the Bernouilli distribution with pmf P[X = x] = p x (1 -p) 1-x ; x =0, 1 (A.1) If, for instance, there are four anodes in a batch of Section 1 and only one among them is of acceptable quality (bad batch!), 3/4 is the probability of any of the anodes being defective, 0.01)/(0.3489-0.3024) = 0.019 is about 8% higher than 0.0176 obtained via numerical integration.Alternatively, the data may be correlated via a properly selected regression.In the case discussed here, inspection of data suggests the semi-linearized form probability of a variable or an event P[E 2 /E 1 ] conditional probability of event E 2 occurring upon the occurrence of event E 1 p single -event success probability pmf probability mass function q single-event failure probability S selectivity u "dummy" integration variable UB unbiased Z standard normal (Gaussian) variate, z its numerical value μ mean of the normal distribution σ 2 variance of the normal distribution Γ(…) gamma function of its argument Φ standard normal probability distribution function of variate Z Ψ(…) auxiliary function [Eq.( 4)]

P
squares fitting (e.g.,Neter et al, 1990) yields a = -10.00894;b= 17.74349; c = 2.836044; d = 6.231738 with coefficients of determination of 0.991 and 0.999, respectively, and I 0.33 (8;5) estimates 0.017598 and 0.017597.Disagreement only in the sixth decimal is a fortuitous finding, inasmuch as such closeness is not guaranteed, in general.The fact, that the pre-integral coefficient -carrying factorials and the Ψ -function often involve the multiplication of very large and very small numbers, might be viewed by some analysts as an incentive for preferring interpolation or regression methods.denoting the number of successes; k its numerical value M, N random variables, denoting the number of failures; m, n their numerical value, respectively ML maximum likelihood N(0;1) standardized random normal distribution with zero mean and unit variance NBD negative binomial distribution P[…]

Table 2 .
Application of the Negative Binomial/Pascal Distribution in Probability Theory to Electrochemical Processes 33 chapter.The mode (the value of the most frequently occurring random variable) does not exist when k =1 (the case of geometric distribution) and k = 0 (meaningless in NBD context).Fundamental parameters of the negative binomial distribution (NBD) followed in this www.intechopen.com

Table 3
Table3contains selected numbers of exact failureprobabilities, and cumulative probabilities that no more than two failures will be observed at the indicated values of k and p.
. Failure/success probabilities in Section 3.1 at selected values of the single-event success probability p, success number k (an exact value of the random variable K of the number of substandard batches), and failure number n (an exact value of the random variable N of batches of acceptable quality).(*): geometric distribution

Table 9 .
Selected "not more than" numbers of acceptable baths in Section 3.4 prior to the appearance of the first bath of unacceptable quality at a small, a medium, and a high success probability.Bath numbers are rounded up or down to the nearest integer.Application of the Negative Binomial/Pascal Distribution in Probability Theory to Electrochemical Processes 41 www.intechopen.com

Table 14 .
Comparison of individual probabilities in Section 7 via NBD and standard normal distribution theory