By Jordan M. Stoyanov

Counterexamples (in the mathematical feel) are strong instruments of mathematical conception. This ebook covers counterexamples from chance conception and stochastic methods. This new elevated version contains many examples and the newest examine effects. the writer is thought of as one of many leading specialists within the box. includes numbers examples.

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**Extra info for Counterexamples in probability**

**Example text**

Then, a natural design wave height would be the maximum wave height reaching the breakwater during its lifetime. However, this value is random and cannot be found. So, the only thing that an engineer can do is to choose this value with a small probability of being exceeded. In order to obtain this probability, it is important to know the probability of exceedances of certain values during a year. Then, if we are concerned with whether the yearly maximum wave height exceeds a given threshold value ho, we have a Bernoulli I experiment.

On the other hand, if the water levels are below given values, the environment can be seriously damaged. 3 Binomial Distribution Suppose now that n Bernoulli experiments are run such that the followiiig conditions hold: 1. The experiments are identical, that is, the probability of success p is the same for all trials. 2. The experiments are independent, that is, the outcome of an experiment has no influence on the outcomes of the others. 2. 2: Examples of probability mass functions of binomial random variables with n = 6 and three values of p.

2. 8) = $625. ) is such that its probability of occurrence during a period of unit duration (normally one year) is a small value p. Assume also that the occurrences of such event in nonoverlapping periods are independent. Then, as time passes, we have a sequence of identical Bernoulli experiments (occurrence or not of the given event). Thus, the time measured in the above units until the first occurrence of this event is the number of experiments until the first occurrence and then it can be considered as a geometric G ( p ) random variable, whose mean is l l p .