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# note 1 - MA2216/ST2131 Probability Notes 1 A Brief...

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MA2216/ST2131 Probability Notes 1 A Brief Historical Account. Three hundred and fifty years ago, it is said, correspondences between two famous French mathematicians, Blaise Pascal and Pierre de Fermat , gave birth to a mathematical theory of probability. The story goes like this: Pascal was approached by a French nobleman, Antoine Gombaud, Chevalier de M´ er´ e, who was interested in gambling. Indeed, Chevalier de M´ er´ e gambled frequently and made a fortune. He bet on a roll of a die that at least one 6 would appear during a total of four rolls. From his own experience, Chevalier de M´ er´ e knew that he was more successful than not with this game of chance. However, if he bet that he would get a total of 12, or a double 6, on twenty-four rolls of two dice, he realized that his old approach to the game resulted in more money. He asked Pascal why his new approach was not as profitable. This problem and others posed by de M´ er´ e led to an exchange of letters between Pascal and Fermat. This took place in 1654. Out of such correspon- dences, some fundamental principles of probability theory were formulated for the first time. The first book on probability was said to be written by a Dutch scientist Christian Huygens in 1657. Understandably, the so-called probability theory developed during that period dealt with primarily problems associated with gambling (i.e., games of chance), and hence, became popular rapidly. Subsequently, Jakob Bernoulli (1654-1705) and Abraham de Moivre (1667-1754) made some major contributions. In 1812, Pierre de Laplace (1749-1827) introduced a host of new ideas and mathematical techniques in one of his books. He also applied probabilistic ideas to many scientific and practical problems. Since then, many have done significant works; among the most important are Chebyshev, Markov, von Mises, Kolmogorov, Paul evy, etc. 1

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An axiomatic approach that forms the basis for the modern theory was presented in 1933 by Kolmogorov, a Russian mathematician, in his celebrated monograph “ Foundations of Probability Theory .” Nowadays, the the- ory of probability has already established itself as an independent branch of mathematical systems. Applications can be found in many practical fields, such as engineering, statistical mechanics (physics), genetics, insurance, and financial mathematics. What to be covered? About 15% on combinatorial probability and 85% on calculus of prob- ability . Course Contents 1. Basic Concepts of Probability Sample space & events; Some basic counting schemes (e.g. permutation, combinations, etc.); Axioms of probability; Some simple properties of probabilities; Union of events; Conditional probability; Independence of events. 2. Discrete Random Variables Random variables; Distribution functions; Discrete density functions; Definition of the expectation (of a function of a random variable); The Bernoulli & binomial random variables; The Poisson random vari- able; Poisson approximation to the binomial distribution; The geometric & negative binomial random variable; The hyper-geometric random vari- able;
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note 1 - MA2216/ST2131 Probability Notes 1 A Brief...

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