Distributions binomial

This distribution has applications in many combinational-type safety and reliability-related problems, and sometimes it is also called a Bernoulli (after its founder Jakob Bernoulli [1654-1705] distribution [1]). 

This probability distribution becomes very helpful in situations where one is concerned with the probabilities of outcome such as the number of failures in a sequence of n trials. It is to be noted that for binomial distribution, each trial has two possible outcomes (e.g., success and failure), and the probability of each trial remains unchanged or constant. 

X is the number of failures in n trials. q is the single trial probability of failure. p is the single trial probability of success. 

The binomial distribution forms the basis for quality control in mass production manufacturing. Consider a large batch of parts having 5% that are defective and 95% that are satisfactory. The probability that the first part taken in a sample of two parts is satisfactory will be 0.95 (call this 5). The probability that the first part is defective will be 0.005 (call this d). The probability that both parts will be satisfactory will be s = (0.95) while the probability that both are defective will be d = (0.05). There are two other possibilities the first may be satisfactory while the second is defective, or the first may be defective and the second satisfactory. Each of these will have a probability of sd) = (0.95)(0.05). Thus, the probability of one satisfactory and one defective = 2sd = 2(0.95)(0.05). These results for a sample of two from a large batch (population) with 5% defective may be summarized as follows  

Consider next, a sample of three from a population with 5% defee-tive drawn in sequenee 1,2, 3. These results are summarized below  

It should be noted that the eoefficients of the s d and scP- terms are permutations of three items at a time with two that are the same  

The probabilities in each of the three cases considered above are  

These are seen to correspond to the binomial expansion of (5 + d) with the exponent n equal to the number of items in the sample. The possibilities for a sample of 10 items taken from a population with 5% defective are as follows  

This distribution is also known as the Bernoulli distribution, after Jakob Bernoulli (1654-1705) [1]. The binomial probability density fimction,/(x), is defined by 

X = number of nonoccurrences (e.g., failures) in m frials p = single frial probability of occurrence (e.g., success) q = single frial probabilify of nonoccurrence (e.g., failure) The cumulative distribution function is given by 

F x) = cumulative distribution function or the probability of x or fewer nonoccurrences (e.g., failures) in m trials Using Equations (2.26) and (2.37), the expected value or the mean of fhe distribution is 

A simple example of a discrete probability distribution is the process by which a single participant is assigned the active treatment when the event active treatment is equally likely as the event placebo treatment. This random process is like a coin toss with a perfectly fair coin. If the random variable, X, takes the value of 1 if active treatment is randomly assigned and 0 if the placebo treatment is randomly assigned, the probability distribution function can be described as follows  

This probability distribution function has the characteristics defined previously  

The first probability distribution function that we discuss in detail is the binomial distribution, which is used to calculate the probability of observing x number of successes out of rt observations. As the random variable of Interest, the 

The probability of observing x successes out of n observations under these conditions (called a Bernoulli process) can be expressed as  

The left part of this expression can be read as the probability of the random variable, X, taking on a particular value of x, given parameters p and n. The quantity (1 - p) is the probability of failure for any trial. The notation C is shorthand to represent the number of combinations of taking x successes out of n observations when ordering is not important. This quantity can be calculated as  

If the possibility of success in any one trial is p, then the possibility of V successes in n trials is given by the binomial distribution  

If we repeat the whole set of n trials many times, the expected mean number of successes is 

Let N be the total number of events independent of each other, p be the probability of success, and x be the number of successful events out of N. Then 1 — p is the probability of failure and N — x is the number of events that fail. The probability that exactly x events will succeed from the group of N is 

The binomial distribution function is not continuous hence, to calculate the average (mean value of x), we have to use operator summation instead of integration  

Equations 2.36 and 2.38 are called Stirling s formula. We will use them both interchangeably. 

Consider p as the probability an event will occur in a single trial (success) and q as the probability this same event will not occur in a single trial (failure). Of course, p - -q =.  

The mean and the variance of a random variable X that is binomially distributed can be determined as follows. First, consider a new, auxiliary variable t.  

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To predict the properties of a population on the basis of a sample, it is necessary to know something about the population s expected distribution around its central value. The distribution of a population can be represented by plotting the frequency of occurrence of individual values as a function of the values themselves. Such plots are called prohahility distrihutions. Unfortunately, we are rarely able to calculate the exact probability distribution for a chemical system. In fact, the probability distribution can take any shape, depending on the nature of the chemical system being investigated. Fortunately many chemical systems display one of several common probability distributions. Two of these distributions, the binomial distribution and the normal distribution, are discussed next. 

Binomial Distribution The binomial distribution describes a population in which the values are the number of times a particular outcome occurs during a fixed number of trials. Mathematically, the binomial distribution is given as... 

A binomial distribution has well-defined measures of central tendency and spread. The true mean value, for example, is given as... 

The binomial distribution describes a population whose members have only certain, discrete values. A good example of a population obeying the binomial distribution is the sampling of homogeneous materials. As shown in Example 4.10, the binomial distribution can be used to calculate the probability of finding a particular isotope in a molecule. 

The probability of finding an atom of in cholesterol follows a binomial distribution, where X is the sought for frequency of occurrence of atoms, N is the number of C atoms in a molecule of cholesterol, and p is the probability of finding an atom of... 

A portion of the binomial distribution for atoms of in cholesterol is shown in Figure 4.5. Note in particular that there is little probability of finding more than two atoms of in any molecule of cholesterol. 

Portion of the binomial distribution for the number of naturally occurring atoms in a molecule of cholesterol. 

As a starting point, let s assume that our target population consists of two types of particles. Particles of type A contain analyte at a fixed concentration, and type B particles contain no analyte. If the two types of particles are randomly distributed, then a sample drawn from the population will follow the binomial distribution. If we collect a sample containing n particles, the expected number of particles containing analyte, ti, is... 

Few populations, however, meet the conditions for a true binomial distribution. Real populations normally contain more than two types of particles, with the analyte present at several levels of concentration. Nevertheless, many well-mixed populations, in which the population s composition is homogeneous on the scale at which we sample, approximate binomial sampling statistics. Under these conditions the following relationship between the mass of a randomly collected grab sample, m, and the percent relative standard deviation for sampling, R, is often valid. ... 

Figure A6.1 and Table A6.1 show how a solute s distribution changes during the first four steps of a countercurrent extraction. Now we consider how these results can be generalized to give the distribution of a solute in any tube, at any step during the extraction. You may recognize the pattern of entries in Table A6.1 as following the binomial distribution...

Furthermore, when both np and nq are greater than 5, the binomial distribution is closely approximated by the normal distribution, and the probability tables in Appendix lA can be used to determine the location of the solute and its recovery. 

The binomial distribution function is one of the most fundamental equations in statistics and finds several applications in this volume. To be sure that we appreciate its significance, we make the following observations about the plausibility of Eq. (1.21) ... 

Also, in many apphcations involving count data, the normal distribution can be used as a close approximation. In particular, the approximation is quite close for the binomial distribution within certain guidelines. 

Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probabinty of obseivdng a specified number of successes x in n trials is defined by the binomial distribution. The sequence of outcomes is called a Bernoulli process, Nomenclature n = total number of trials X = number of successes in n trials p = probability of obseivdng a success on any one trial p = x/n, the proportion of successes in n triails Probability Law... 

Nature In an experiment in which one samples from a relatively small group of items, each of which is classified in one of two categories, A or B, the hypergeometric distribution can be defined. One example is the probabihty of drawing two red and two black cards from a deck of cards. The hypergeometric distribution is the analog of the binomial distribution when successive trials are not independent, i.e., when the total group of items is not infinite. This happens when the drawn items are not replaced. 

When the value of p is very close to zero in Eq. (9-77), so that the occurrence of the event is rare, the binomial distribution can be approximated by the Poisson distribution with X = np when n > 50 while npi < 5. 

Any data set that consists of discrete classification into outcomes or descriptors is treated with a binomial (two outcomes) or multinomial (tliree or more outcomes) likelihood function. For example, if we have y successes from n experiments, e.g., y heads from n tosses of a coin or y green balls from a barrel filled with red and green balls in unknown proportions, the likelihood function is a binomial distribution ... 

A related problem is to find the probability of M failures or less out of N components. This is found by summing equation 2.4-9 for values less than M as given by equation 2.4-10 which can be used to calculate a one-sided confidence bound over a binomial distribution (Abramowitz and Stegun, p. 960). 

The Poisson distribution follows naturally from the discrete binomial distribution already introduced in the craps and the M-out-of-N problem. As N becomes large, the Poisson distribution approximates the binomial distribution... 

The derivation will not be provided. Suffice it to say that the failures in a time interval may be modeled using the binomial distribution. As these intervals are reduced in size, this goes over to the Poisson distribution and the MTTF is chi-square distributed according to equation 2.9-31, where = 2 A N T and the degrees of freedom,/= 2(M+i). 

The cumulative binomial distribution is given by equation 2.5-33, where M is the number of f ailures out of items each having a probability of failure p. This can be worked backH tirLh lo find tlic implied value of p for a specified P(M, p,... 

This table indicates that if a beta function prior is convoluted with a binomially distributed update, the combination (the posterior) also is beta distributed. 

A lrLL[uently encountered problem requires estimating a failure probability based on the number of failures, M, in N tests. These updates are assumed to be binomially distributed (equation 2.4-10) as p r N). Conjugate to the binomial distribution is the beta prior (equation 2.6-20), where / IS the probability of failure. 

In the introduction to this section, two differences between "classical" and Bayes statistics were mentioned. One of these was the Bayes treatment of failure rate and demand probttbility as random variables. This subsection provides a simple illustration of a Bayes treatment for calculating the confidence interval for demand probability. The direct approach taken here uses the binomial distribution (equation 2.4-7) for the probability density function (pdf). If p is the probability of failure on demand, then the confidence nr that p is less than p is given by equation 2.6-30. 

This, more physical model that visualizes failure to result from random "shocks," was specialized from the more general model of Marshall and Olkin (1967) by Vesely (1977) for sparse data for the ATWS problem. It treats these shocks as binomially distributed with parameters m and p (equation 2.4-9). The BFR model like the MGL and BPM models distinguish the number of multiple unit failures in a system with more than two units, from the Beta Factor model,... 

A number of issues arise in using the available data to estimate (he rates of location-dependent fire occurrence. These include the possible reduction in the frequency of fires due to increased awareness. Apostolakis and Kazarians (1980) use the data of Table 5.2-1 and Bayesian analysis to obtain the results in Table 5.2-2 using conjugate priors (Section 2.6.2), Since the data of Table 5.2-1 are binomially distributed, a gamma prior is used, with a and P being the parameters of the gamma prior as presented inspection 2.6.3.2. For example, in the cable- spreading room fromTable 5.2-2, the values of a and p (0.182 and 0.96) yield a mean frequency of 0.21, while the posterior distribution a and p (2.182 and 302,26) yields a mean frequency of 0.0072. 

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