Probability and Statistics
Volume I
Published on 19. December 1985
Book
Hardback
362 pages
978-0-387-96067-8 (ISBN)
Description
Who is a probabilist? Someone who knows the odds of drawing an ace of diamonds, or the waiting time at a ticket office? A mathematician who uses a special vocabulary? And who is a statistician? Someone who is capable of determining whether tobacco encourages cancer, or the number of votes which a certain candidate will poll at the next elections? Each one perceives the link: chance. The probabilist, guided by his intuition of poker or queues, constructs an abstract model; having fixed the mathematical framework, he calmly follows through his logical reasoning, which sometimes him very far from his starting point. The statistician takes works on solid ground. When a doctor asks him if it is worth using a new drug, he uses the tools of a probabilist. However he must reach a decision, the least harmful option, based on analyzing the doctor's observations, while taking into account the various risks involved. In short, a probabilist keeps his hands clean while dreaming of models, while a statistician must dirty his hands while working with concrete facts. Relations between the two have often been difficult; but the barriers to their dialogue are broken down by the interest in the concrete to supplement theoretical dreams or in complicated models to describe various phenomena. However, few students have a chance to overcome these barriers.
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Content
1 Censuses.- 1.1. Census of Two Qualitative Characteristics.- 1.2. A Census of Quantitative Characteristics.- 1.3. First Definitions of Discrete Probabilities.- 1.4. Pairs of R.V.'s and Correspondence Analysis.- 2 Heads or Tails. Quality Control.- 2.1. Repetition of n Independent Experiments.- 2.2. A Bernoulli Sample.- 2.3. Estimation.- 2.4. Tests, Confidence Intervals for a Bernoulli Sample, and Quality Control.- 2.5. Observations of Indeterminate Duration.- 3 Probabilistic Vocabulary of Measure Theory. Inventory of the Most Useful Tools.- 3.1. Probabilistic Models.- 3.2. Integration.- 3.3. The Distribution of a Measurable Function.- 3.4. Convergence in Distribution.- 4 Independence: Statistics Based on the Observation of a Sample.- 4.1. Sequence of n Observations-Product Measure Spaces.- 4.2. Independence.- 4.3. Distribution of the Sum of Independent Random Vectors.- 4.4. A Sample from a Distribution and Estimation of this Distribution.- 4.5. Nonparametric Tests.- 5 Gaussian Samples, Regression, and Analysis of Variance.- 5.1. Gaussian Samples.- 5.2. Gaussian Random Vectors.- 5.3. Central Limit Theorem on ?k.- 5.4. The X2 Test.- 5.5. Regression.- 6 Conditional Expectation, Markov Chains, Information.- 6.1. Approximation in the Least Squares Sense by Functions of an Observation.- 6.2. Conditional Expectation-Extensions.- 6.3. Markov Chains.- 6.4. Information Carried by One Distribution on Another.- 7 Dominated Statistical Models and Estimation.- 7.1. Dominated Statistical Models.- 7.2. Dissimilarity in a Dominated Model.- 7.3. Likelihood.- 8 Statistical Decisions.- 8.1. Decisions.- 8.2. Bayesian Statistics.- 8.3. Optimality Properties of Some Likelihood Ratio Tests.- 8.4. Invariance.- Notation and Conventions.