## Introduction to ProbabilityProbability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two problems from games of chance. Problems like those Pascal and Fermat solved continued to influence such early researchers as Huygens, Bernoulli and DeMoivre in establishing a mathematical theory of probability. Today, probability theory is a well-established branch of mathematics that finds applications in every area of scholarly activity from music to physics, and in daily experience from weather prediction to predicting the risks of new medical treatments. This text is designed for an introductory probability course taken by sophomores, juniors and seniors in mathematics, the physical and social sciences, engineering and computer science. It presents a thorough treatment of probability ideas and techniques necessary for a form understanding of the subject. The text can be used in a variety of course lengths, levels, and areas of emphasis. For use in a standard one-term course, in which both discrete and continuous probability is covered, students should have taken as a prerequisite two terms of calculus, including an introduction to multiple integrals. In order to cover Chapter 11, which contains material on Markov chains, some knowledge of matrix theory is necessary. The text can also be used in a discrete probability course. The material has been organized in such a way that the discrete and continuous probability discussions are presented in a separate, but parallel, manner. This organization dispels an overly rigorous or formal view of probability and offers some strong pedagogical value in that the discrete discussions can sometimes serve to motivate the more abstract continuous probability discussions. For use in a discrete probability course, students should have taken one term of calculus as a prerequisite. Very little computing background is assumed or necessary in order to obtain full benefits from the use of the computing material and examples in the text. All of the programs that are used in the text have been written in each of the languages TrueBASIC, Maple and Mathematica. |

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### Conteúdo

Discrete Probability Distributions | 1 |

Continuous Probability Densities | 41 |

Combinatorics | 75 |

Conditional Probability | 133 |

Distributions and Densities | 183 |

Expected Value and Variance | 225 |

Sums of Random Variables | 285 |

Law of Large Numbers | 305 |

Central Limit Theorem | 325 |

Generating Functions | 365 |

Markov Chains | 405 |

Random Walks | 471 |

Appendices | 499 |

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approximately assign assume balls bar graph Bernoulli trials binomial distribution branching process calculate called cards Central Limit Theorem Chebyshev's Inequality choose chosen at random coin is tossed consider continuous random variable cumulative distribution function deck defined denote density function dice discrete random variables distribution with parameter dollars equal equation ergodic chain estimate event Example Exercise expected number expected value experiment exponential density fair coin Find the probability finite formula fx(x gambler given independent random variables independent trials process interval 0,1 Large Numbers Law of Large Markov chain Mathematics mean normally distributed number of heads obtain occurs offspring pair Pascal percent permutation play player Poisson distribution possible outcomes probability vector problem proof queue random numbers random walk real number rolls sample space Show simulation subset Suppose Table transition matrix variance a2 Write a program