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Almost impossible integrals sums and series how to#
Almost impossible integrals sums and series professional#
Almost impossible integrals sums and series series#

In the foreword to his treatise, Huygens wrote:

Almost impossible integrals sums and series how to#

The book extended the concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players), and can be seen as the first successful attempt at laying down the foundations of the theory of probability. Huygens published his treatise in 1657, (see Huygens (1657)) " De ratiociniis in ludo aleæ" on probability theory just after visiting Paris. In Dutch mathematician Christiaan Huygens' book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. They only informed a small circle of mutual scientific friends in Paris about it. They were very pleased by the fact that they had found essentially the same solution, and this in turn made them absolutely convinced that they had solved the problem conclusively however, they did not publish their findings.

This principle seemed to have come naturally to both of them. The principle is that the value of a future gain should be directly proportional to the chance of getting it. They solved the problem in different computational ways, but their results were identical because their computations were based on the same fundamental principle. Soon enough, they both independently came up with a solution.

Almost impossible integrals sums and series series#

He began to discuss the problem in the famous series of letters to Pierre de Fermat. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all. Méré claimed that this problem couldn't be solved and that it showed just how flawed mathematics was when it came to its application to the real world. Many conflicting proposals and solutions had been suggested over the years when it was posed to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré in 1654. This problem had been debated for centuries. The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes in a fair way between two players, who have to end their game before it is properly finished.

5.3 Relationship with characteristic function.

5.2 Expectations under convergence of random variables.

4 Expected values of common distributions.

3.4 Arbitrary real-valued random variables.

3.2 Random variables with countably many outcomes.

3.1 Random variables with finitely many outcomes.

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Almost impossible integrals sums and series professional#

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