Probability theory independent events

According to the Classical Nucleation Theory, the repetitive formation of nuclei in a metastable liquid can be considered as a sequence of independent events and the distribution of metastable lifetimes shows an exponential decrease (see also Takahashi et This implies that the density probability function f(t) of the nucleation event is ... 

Since both k and all the y/ s can be related to probabilities (refer back to Section 14.1.1), the orbital approximation defines a total probability as a product of individual probabilities. This is only true in probability theory if the individual events (y/ s) are independent, such as each flipping of a coin to get "heads" or "tails" is an independent event. With respect to electrons, this means that the probability that electron 1 will exist at a certain position in space is completely independent of the positions of electrons 2, 3,4, etc. The electrons movements are therefore not correlated, a severe approximation (see the discussion of electron correlation given later). This is therefore called an independent electron theory. 

An important feature of the random-walk theory is that in a sequence of trials each trial is independent of the others. Markov advanced the theory by generalizing the condition that the outcome of any trial may depend on the outcome of the preceding trial that is, the probability of an event is conditioned by the previous event. This idea clearly fits the description of the configuration of a polymer chain. 

A component failure is an independent failure is when the failure is solely due to an inherent failure mode of the component, and is not caused by the failure of a different component or an external event. In probability theory, events are independent when the outcome of one event does not influence the... 

We assume that the probability that a randomly chosen molecule has a particular velocity is independent of the probability that it has a particular position. It is a fact of probability theory that the number of ways of accomplishing two independent events is the product of the number of ways of accomplishing each event. The thermodynamic probability is therefore the product of two factors, one for the coordinates and one for the velocities ... 

The two can be related using a concept called a Poisson process. From the Wikipedia s article on the Poisson Process In probability theory, a Poisson process is a stochastic process which counts the number of events and the time that these events occur in a given time interval. The time between each pair of consecutive events has an exponential distribution with parameter G the parameter is the occurrence rate per unit time] and each of these inter-arrival times is. .. independent of other... 

In the theory of probability the term correlation is normally applied to two random variables, in which case correlation means that the average of the product of two random variables X and Y is the product of their averages, i.e., X-Y)=(,XXY). Two independent random variables are necessarily uncorrelated. The reverse is usually not true. However, when the term correlation applies to events rather than to random variables, it becomes equivalent to dependence between the events. 

Statistical theory teaches that under the assumption that the population means of the two groups are the same (i.e. if Hq is true), the distribution of variable T depends only on the sample size but not on the value of the common mean or on the measurements population variance and thus can be tabulated independently of the particulars of any given experiment. This is the so-called Student s f-distribution. Using tables of the f-distribution, we can calculate the probability that a variable T calculated as above assumes a value greater or equal to 4.7, the value obtained in our example, given that H0 is true. This probability is <0.0001. Thus, if H0 is true, the result obtained in our experiment is extremely unlikely, although not impossible. We are forced to choose between two possible explanations to this. One is that a very unlikely event occurred. The second is that the result of our experiment is not a fluke, rather, the difference Mb — Ma is a positive number, sufficiently large to make the probability of this outcome a likely event. We elect the latter explanation and reject H0 in favor of the alternative hypothesis Hx. 

Conditional probability, and the related topic of independence, give probability a unique character. As aheady mentioned, mathematical probability can be presented as a topic in measure theory and general integration. The idea of conditional probabihty distinguishes it from the rest of mathematical analysis. Logically, independence is a consequence of conditional probability, but the exposition to follow attempts an intuitive explanation of independence before introducing conditional events. 

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