Bernoulli process

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... 

More complex schemes have been proposed, such as second-order Markov chains with four independent parameters (corresponding to a copolymerization with penultimate effect, that is, an effect of the penultimate member of the growing chain), the nonsymmetric Bernoulli or Markov chains, or even non-Maikov models a few of these will be examined in a later section. Verification of these models calls for the knowledge of the distribution of sequences that become longer, the more complex the proposed mechanism. Considering only Bernoulli and Markov processes it may be said that at the dyad level all models fit the experimental data and hence none can be verified at the triad level the Bernoulli process can be verified or rejected, while all Markov processes fit at the tetrad level the validity of a first-order Markov chain can be confirmed, at the pentad level that of a second-order Maikov chain, and so on (10). 

A temperature influence on the stereocontrol of polymerization reactions was postulated by Huggins (17) as early as 1944 and backed by the experimental results of Fordham (16) and Bovey (18). A quantitative description of the temperature dependence was tried by Fordham (16) on the basis of the transition state theory. For a Bernoulli process, he obtained ... 

Bernoulli process (p + p = 0.909). Similar results were also... 

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

The geometric distribution indicates the probability of conducting x trials to obtain a success in an experiment in which there are only two possible outcomes. Like the binomial distribution, this is another Bernoulli process. Each trial is assumed to be independent, and the probability of observing a success is constant over all trials, denoted p. The probability distribution for the geometric distribution [2] is... 

Using the above two approximations, we can model the bit errors as a Bernoulli process i.e. we have a sequence of independent bits, where all the bits have the same probability of error xBERm-... 

The probabilities and rate constants for isotactic and syndiotactic addition are different in a Bernoulli process with respect to diads ... 

Observations come from a Bernoulli process when they are from a sequence of independent Bernoulli trials. For Bernoulli trials, each trial has two possible outcomes which we label "success" and "failure." The probability of success, tt, remains constant over all the trials. The binomial n, n) distribution arises when Y is the number of "successes" in a sequence of n Bernoulli trials with success probability tt. 

Sometimes the random variable we are interested in is the time we have to wait until some event occurs. In this section we look at four distributions for waiting time, two discrete distributions associated with Bernoulli processes, and two continuous distributions associated with the Poisson process. In each situation the first distribution is a special case of the second. 

Many commonly used distributions are members of the one-dimensional exponential family. These include the binomial n, n) and Poisson fi) distributions for count data, the geometric ir) and negative binomial r, tt) distributions for waiting time in a Bernoulli process, the exponential ) and gamma n, A) distributions for waiting times in a Poisson process, and the normal p, o ) where [Pg.89]

The results of the studies.discussed in Section II,C permit calculations to be made of the time required for the flame to spread to the entire propellant surface. Once this phase of the motor-ignition process has been completed, the time required to fill the combustion chamber and establish the steady-state operating conditions must be computed. This can be done by the formal solution of Eq. (7). Because this equation is a Bernoulli type of nonlinear equation, the formal solution becomes... 

In many cases the equations for the longer sequences (tetrads, pentads, etc.) are markedly simplified if they are normalized not on the totality of the sequences but only on those centered on the same letter or group of letters. For example, the reduced frequency of the mmm tetrad with respect to all the m centered tetrads or that of the mmrr pentad with respect to all the mr centered pentads is expressed by quadratic expressions for both the Bernoulli and Markov processes. 

The statistical treatment of a hemiisotactic polymer can be made on the basis of a single parameter a the corresponding formulas are reported in Table 4, last column. For extreme values of a the polymer is no longer hemiisotactic but syndiotactic (for a = 0) or isotactic (for a = 1). The particular distribution existing in the hemiisotactic polymer is not reproducible with either the Bernoulli or the Markov processes expressed in m/r terms. 

For syndiotactic polypropylene the symmetric Bernoulli trial, expressed in m and r dyads, is quite adequate for the representation of experimental data, and agrees with the stereochemical control being exerted by the growing chain end (145, 409). In its turn, atactic polypropylene is considered as a mixture of the products of two superposed processes, of the type discussed for isotactic and syndiotactic polymers, and is described by a simplified two-state model (145). 

Box 18.1 Deterministic and Random Processes Bernoulli Coefficients Normal Distributions... 

The concatenation is performed without constraint on the successive symbols so that the motion on the repeller corresponds to a Bernoulli random process. The regions around the fundamental periodic orbits are successively visited in a random fashion without memory of the previous fundamental periodic orbit visited. As a consequence, the periodic orbits proliferate exponentially with their period, as described by (2.17). The topological entropy per symbol is equal to htop = In M. 

A very useful equation to deal with phenomena associated with the flow of fluids is the Bernoulli equation. It can be used to analyse fluid flow along a streamline from a point 1 to a point 2 assuming that the flow is steady, the process is adiabatic and that frictional forces between the fluid and the tube are negligible. Various forms of the equation appear in textbooks on fluid mechanics and physics. A statement in differential form can be obtained ... 

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