An Elementary Introduction to the Theory of Probability (Dover Books on Mathematics) by B. V. Gnedenko (PDF)

Description

This compact volume equips the reader with all the facts and principles essential to a fundamental understanding of the theory of probability. It is an introduction, no more: throughout the book the authors discuss the theory of probability for situations having only a finite number of possibilities, and the mathematics employed is held to the elementary level. But within its purposely restricted range it is extremely thorough, well organized, and absolutely authoritative. It is the only English translation of the latest revised Russian edition; and it is the only current translation on the market that has been checked and approved by Gnedenko himself.After explaining in simple terms the meaning of the concept of probability and the means by which an event is declared to be in practice, impossible, the authors take up the processes involved in the calculation of probabilities. They survey the rules for addition and multiplication of probabilities, the concept of conditional probability, the formula for total probability, Bayes’s formula, Bernoulli’s scheme and theorem, the concepts of random variables, insufficiency of the mean value for the characterization of a random variable, methods of measuring the variance of a random variable, theorems on the standard deviation, the Chebyshev inequality, normal laws of distribution, distribution curves, properties of normal distribution curves, and related topics.The book is unique in that, while there are several high school and college textbooks available on this subject, there is no other popular treatment for the layman that contains quite the same material presented with the same degree of clarity and authenticity. Anyone who desires a fundamental grasp of this increasingly important subject cannot do better than to start with this book. New preface for Dover edition by B. V. Gnedenko.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I like the book so far. The exposition isn’t fluffy or terse. There is an arithmetic error on page 24 where .1 * .2 * .15 is stated to equal 0.0003It looks like there is a second error on page 51; “values of the number k which which lie at a distance not more than” should read “values of the number k which which lie at a distance more than”

⭐This brief text, which was written for high school students in the Soviet Union following World War II, is an illuminating introduction to probability theory that does not require a foundation in calculus. The authors develop the theory by generalizing from examples, most of which are taken from military or industrial applications. This gives the reader insight into how mathematicians develop theorems by abstracting from problems arising in the real world. The theorems are proved rigorously except in the final chapter on normal distributions. Formal proofs about normal distributions require advanced mathematics not familiar to the intended audience.Probability theory is developed in the first section of the text. The authors define probability. They explain the addition rule and how it simplifies when events are mutually exclusive. Likewise, after they obtain the multiplication rule in terms of conditional probabilities, they explain how it simplifies when events are mutually independent. The authors discuss Bayes’ formula for the probability of a hypothesis given that a given event has been observed using several examples. They then prove Bernoulli’s formula for the most probable number of occurrences of an event when there are a large number of trials.The second section of the text is on random variables. The authors discuss laws of distribution, mean values, variance and standard of deviation, and how these quantities are used to measure the dispersion of a random variable. Their development culminates in Chebyshev’s law of large numbers. In the final chapter on normal distributions, the authors informally discuss their properties and show how they can be used to solve problems.In a brief conclusion, the authors discuss other developments in probability theory that are beyond the scope of this text.This text is an excellent introduction to probability theory. I recommend it highly for the insights it offers. However, it does not contain exercises. To learn mathematics, one must solve problems. Therefore, I suggest that you read this text in conjunction with a problem book on probability or a text on probability that does contain exercises such as Samuel Goldberg’s

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⭐Even though the fifth edition of this book was published in 1961, this book still gives a useful and brief introduction to probability. However, if you’re going to buy a book to learn statistics or probability, I would suggest a more recent book. They may not be as brief and concise as this, but newer books would be more up to date. One interesting thing about the Gnedenko/Khinchin book is their examples and problems, which involve things such as the production of artillery shells, or hitting targets with cannons. It is a very welcome change from the traditional, but obvious examples that use decks of cards or dice. Also, it says things about the audience for whom this book was originally meant, and the relevant topics of the time of the cold war. Still, I suggest a more recent text.

⭐This book explains concepts simply and clearly. It is the exact opposite of a typical American math book which is filled with useless formalism and endlessly repeated examples. This book conveys the core concepts of probability quickly and seemingly effortlessly.

⭐本書は120頁と薄くはあるけれども、確率論で重要なBernoulliの定理、Chebyishevの不等式、大数の法則、正規分布を含んでおり、それらを具体的な例でもって説明し初等的な程度の数学で表現してある。確率論に貢献の大きいロシアのGnedenkoとKhinchinによる確率論入門である。高校生が確率論に興味を持って、きちんとした数学的理解をしようと思ったならば、本書を最初に読んでみるのがよかろう。それから例えば大部ではあるが懇切丁寧なFellerの本などに進むことができる。

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