Probability model Bernoulli distribution

Model 1 Zi,..., are independent and identically distributed observations from a Bernoulli distribution with probability p. These Bernoulli random variables take the value 0 and 1. 

Using the statistical terminology, a CSBS can be modeled by the n-dimensional Bernoulli distribution. As is well known (see, e.g., (10)), this distribution consists of n random variables Xj,. .., x , which rally take two possible values, 0 or 1, with probabilities... 

Setting n=, that is, only a single trial occurs, we get the Bernoulli distribution. It models the probability of a single trial given two possibilities. Historically, this distribution was first proposed and then generalised to the binomial distribution. 

Poisson Distribution The Poisson distribution can serve as a model for a number of different types of experiments. For example, when the number of opportunities for the event is very large but the probability that the event occurs in any specific instance is very small, the number of occurrences of a rare event can sometimes be modeled by the Poisson distribution. Moreover, the occurrences are i.i.d. Bernoulli trials. Other examples are the number of earthquakes and the number of leukemia cases. 

The theoretical model simulates the reaction scheme of the intermittent propagation of Fig. 7 on the basis of a statistical distribution of the polymerization activity onto all molecules (C ) present in the reactor. In other words, the possibility to become an active species is again distributed newly after each insertion step, because the concentration of the different alkyl chains is changed after each insertion step. Figure 14 shows the binomial distribution formula or, more precisely, the Bernoulli scheme for two incompatible events. In this formula, a is the probabiUty for the event, 1—a the non-probability for the event, and v the number of times that the event occurs. 

Temporal Earthquake Uncertainty The temporal uncertainty of earthquakes is included in PSHA by considering the distribution of their occurrence in time, which can be modeled by random models, such as Bernoulli, Poisson, and Markov processes. In this manner, it is possible to calculate the probability of the number of occurrences over a certain time interval. The occurrence of an earthquake for a certain time interval is usually modeled as a Poisson process that describes the number of occurrences of an event (not only earthquakes but floods and other natural disasters as well) during a specified time interval and provides also the return period of such event. The main assumptions of Poisson model are as follows (Kramer 1996) ... 

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