Bernoulli trial process

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). 

The discrete equivalent of the Poisson process is related to a series of Bernoulli trials, in which case the individual trials (e.g., coin tosses) can be assigned to individual discrete moments (i.e., to the serial number of the toss, n). It is clear that the process X(n) has a B n, p) binomial distribution - the distribution characteristic of the number of heads turning up in a series of n tosses. 

Some probability distribution functions occur frequently in nature, and have simple mathematical expressions. Two of the most useful ones are the binomial and multinomial distribution functions. These will be the basis for our development of the concept of entropy in Chapter 2. The binomial distribution describes processes in which each independent elementary event has two mutually exclusive outcomes such as heads/tails, yes/no, up/down, or occu-pied/vacant. Independent trials with two such possible outcomes are called Bernoulli trials. Let s label the two possible outcomes 9 and J. Let the probability of be p. Then the probability of J is 1 - p. We choose composite events that are pairs of Bernoulli trials. The probability of followed by is P0j = p(l - p). The probabilities of the four possible composite events are... 

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. 

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... 

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... 

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