{"loaderData":{"0-0-57":{"chapterData":{"_id":"66be7711be903046a22f2b9f","book_id":"66bd2adfbe903046a22f1be8","chapter_name":"Normal, Poisson, and Related Probability Laws","public_visibility":true,"book_slug":"probability-theory-and-its-applications","slug":"normal-poisson--and-related-probability-laws","regular_authors":"","express_authors":"","regular_content":"

In applied probability theory the binomial, normal, and Poisson probability laws play a central role. In this chapter we discuss the reasons for the importance of the normal and Poisson probability laws.

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Modern Probability Theory and Its Applications by Emanuel Parzen is a groundbreaking text that has profoundly influenced generations of statisticians and non-statisticians. What sets this book apart is its unique blend of mathematical rigor and practical applicability. Engineers praise it as an \"engineering book\" disguised as a mathematics text, making it exceptionally readable and relevant. The content is both accessible to students and encyclopedic in scope, covering a wide range of topics from basic concepts to advanced probability theory.

For students seeking a thorough understanding of probability theory and its practical applications, and for professors aiming to provide a comprehensive, engaging course in probability, Parzen's Modern Probability Theory and Its Applications remains an unparalleled resource that has stood the test of time.

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The purpose of this chapter is to discuss the nature of probability theory. In Section 1 we point out the existence of a certain body of phenomena that may be called random. In Section 2 we state the view, which is adopted in this book, that probability theory is the study of mathematical models of random phenomena. The language and notions that are used to formulate mathematical models are discussed in Sections 3 to 7.

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Many of the basic concepts of probability theory, as well as a large number of important problems of applied probability theory, may be considered in the context of finite sample description spaces and thus can be studied with a minimum of mathematical technique. In Chapters 2 and 3 only finite sample description spaces are considered. In this chapter we further restrict ourselves to finite sample description spaces with equally likely descriptions.

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In this chapter we show how to treat probability problems involving finite sample description spaces, in which the descriptions are not necessarily equally likely, by using the notions of independent and dependent events and trials.

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In the foregoing we have considered mainly random phenomena whose sample description spaces were finite. We next consider random phenomena for which this is not necessarily the case. The simplest example of a random phenomenon whose sample description space is not necessarily finite is one which is numerical valued. The height of waves on a windswept sea, the number of alpha particles emitted from a radioactive source, the number of telephone calls arriving at a switchboard, the velocity of a particle in Brownian motion, the scores of students on an examination, the collar sizes of men, the dress sizes of women, and so on, constitute examples of numerical-valued random phenomena. In this chapter we discuss the notions and techniques used to treat numerical-valued random phenomena.

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It has been emphasized that in order to describe a numerical valued random phenomenon one must specify its probability function \\(P[\\cdot]\\) or, equivalently, its distribution function \\(F(\\cdot)\\) . In the special case in which the random phenomenon obeys a discrete or a continuous probability law its probability function is determined by a knowledge of the probability mass function \\(p(\\cdot)\\) or of the probability density function \\(f(\\cdot)\\) . Thus, to describe a numerical valued random phenomenon, certain functions must be specified. It is desirable to be able to summarize some of the outstanding features of the probability law of a numerical valued random phenomenon by specifying only a few numbers rather than an entire function. Such numbers are provided by the expectation of various functions \\(g(\\cdot)\\) with respect to the probability law of the random phenomenon.

","chapter_express_content":"","public_visibility":true,"book_slug":"probability-theory-and-its-applications","slug":"mean-and-variance-of-a-probability-law","regular_authors":"","express_authors":"","created_at":"2024-08-14T10:38:09.042Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"mean-and-variance-of-a-probability-law","slug":"the-notion-of-an-average","header_name":"The Notion of an Average","type":"header-1","header_data":[],"_id":"66be4caebe903046a22f2909"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"mean-and-variance-of-a-probability-law","slug":"expectation-of-a-function-with-respect-to-a-probability-law","header_name":"Expectation of a Function With Respect to a Probability Law","type":"header-1","header_data":[],"_id":"66be4cd5be903046a22f291d"},{"sequence":3,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"mean-and-variance-of-a-probability-law","slug":"moment-generating-functions","header_name":"Moment-Generating Functions","type":"header-1","header_data":[],"_id":"66be65ebbe903046a22f29da"},{"sequence":4,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"mean-and-variance-of-a-probability-law","slug":"chebyshevs-inequality","header_name":"Chebyshev's Inequality","type":"header-1","header_data":[],"_id":"66be75e9be903046a22f2aea"},{"sequence":5,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"mean-and-variance-of-a-probability-law","slug":"the-law-of-large-numbers-for-independent-repeated-bernoulli-trials","header_name":"The Law of Large Numbers for Independent Repeated Bernoulli Trials","type":"header-1","header_data":[],"_id":"66be76b4be903046a22f2b77"},{"sequence":6,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"mean-and-variance-of-a-probability-law","slug":"more-about-expectation","header_name":"More About Expectation","type":"header-1","header_data":[],"_id":"66be76debe903046a22f2b8b"}]},{"_id":"66be7711be903046a22f2b9f","book_id":"66bd2adfbe903046a22f1be8","chapter_name":"Normal, Poisson, and Related Probability Laws","sequence":6,"chapter_content":"

","chapter_express_content":"","public_visibility":true,"book_slug":"probability-theory-and-its-applications","slug":"normal-poisson--and-related-probability-laws","regular_authors":"","express_authors":"","created_at":"2024-08-14T10:38:09.042Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"normal-poisson--and-related-probability-laws","slug":"the-importance-of-the-normal-probability-law","header_name":"The Importance of the Normal Probability Law","type":"header-1","header_data":[],"_id":"66be7737be903046a22f2bb3"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"normal-poisson--and-related-probability-laws","slug":"the-approximation-of-the-binomial-probability-law-by-the-normal-and-poisson-probability-laws","header_name":"The Approximation of the Binomial Probability Law by the Normal and Poisson Probability Laws","type":"header-1","header_data":[],"_id":"66be7757be903046a22f2bc7"},{"sequence":3,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"normal-poisson--and-related-probability-laws","slug":"the-poisson-probability-law","header_name":"The Poisson Probability Law","type":"header-1","header_data":[],"_id":"66be7779be903046a22f2bdb"},{"sequence":4,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"normal-poisson--and-related-probability-laws","slug":"the-exponential-and-gamma-probability-laws","header_name":"The Exponential and Gamma Probability Laws","type":"header-1","header_data":[],"_id":"66be77a5be903046a22f2bef"},{"sequence":5,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"normal-poisson--and-related-probability-laws","slug":"birth-and-death-processes","header_name":"Birth and Death Processes","type":"header-1","header_data":[],"_id":"66be77d4be903046a22f2c03"}]},{"_id":"66be77f0be903046a22f2c17","book_id":"66bd2adfbe903046a22f1be8","chapter_name":"Random Variables","sequence":7,"chapter_content":"

It has been stressed in the foregoing chapters that the probability of a random event can be discussed only with reference to a sample description space on which a probability function has been defined. However, in many applications of probability theory the terminology of sample description spaces does not explicitly enter (although, as we shall see, the notion is always implicitly present). Rather, many applications of probability theory are based on the notion of a random variable . This chapter gives a rigorous definition of the notion of a random variable and presents the main concepts and techniques used to treat random variables.

","chapter_express_content":"","public_visibility":true,"book_slug":"probability-theory-and-its-applications","slug":"random-variables","regular_authors":"","express_authors":"","created_at":"2024-08-14T10:38:09.042Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"random-variables","slug":"the-notion-of-a-random-variable","header_name":"The Notion of a Random Variable","type":"header-1","header_data":[],"_id":"66be7816be903046a22f2c2b"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"random-variables","slug":"describing-a-random-variable","header_name":"Describing a Random Variable","type":"header-1","header_data":[],"_id":"66be7bffbe903046a22f2c3f"},{"sequence":3,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"random-variables","slug":"an-example-treated-from-the-point-of-view-of-numerical-n-tuple-valued-random-phenomena","header_name":"An Example, Treated From the Point of View of Numerical n-Tuple Valued Random Phenomena","type":"header-1","header_data":[],"_id":"66bf02e1be903046a22f351f"},{"sequence":4,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"random-variables","slug":"the-same-example-treated-from-the-point-of-view-of-random-variables","header_name":"The Same Example Treated From the Point of View of Random Variables","type":"header-1","header_data":[],"_id":"66be7c5dbe903046a22f2c67"},{"sequence":5,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"random-variables","slug":"jointly-distributed-random-variables","header_name":"Jointly Distributed Random Variables","type":"header-1","header_data":[],"_id":"66be7cb9be903046a22f2c7b"},{"sequence":6,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"random-variables","slug":"independent-random-variables","header_name":"Independent Random Variables","type":"header-1","header_data":[],"_id":"66be7ce2be903046a22f2c8f"},{"sequence":7,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"random-variables","slug":"random-samples-randomly-chosen-points-geometrical-probability-and-random-division-of-an-interval","header_name":"Random Samples, Randomly Chosen Points (Geometrical Probability), and Random Division of an Interval","type":"header-1","header_data":[],"_id":"66be7d14be903046a22f2ca3"},{"sequence":8,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"random-variables","slug":"the-probability-law-of-a-function-of-a-random-variable","header_name":"The Probability Law of a Function of a Random Variable","type":"header-1","header_data":[],"_id":"66be7d69be903046a22f2cf6"},{"sequence":9,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"random-variables","slug":"the-probability-law-of-a-function-of-random-variables","header_name":"The Probability Law of a Function of Random Variables","type":"header-1","header_data":[],"_id":"66be7d8cbe903046a22f2d0a"},{"sequence":10,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"random-variables","slug":"the-joint-probability-law-of-functions-of-random-variables","header_name":"The Joint Probability Law of Functions of Random Variables","type":"header-1","header_data":[],"_id":"66be7dafbe903046a22f2d1e"},{"sequence":11,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"random-variables","slug":"conditional-probability-of-an-event-given-a-random-variable--conditional-distributions","header_name":"Conditional Probability of an Event Given a Random Variable. Conditional Distributions","type":"header-1","header_data":[],"_id":"66be7deabe903046a22f2d32"}]},{"_id":"66be8d41be903046a22f2d76","book_id":"66bd2adfbe903046a22f1be8","chapter_name":"Expectation of a Random Variable","sequence":8,"chapter_content":"

In dealing with random variables, it is as important to know their means and variances as it is to know their probability laws. In this chapter we define the notion of the expectation of a random variable and describe the significant role that this notion plays in probability theory.

","chapter_express_content":"","public_visibility":true,"book_slug":"probability-theory-and-its-applications","slug":"expectation-of-a-random-variable","regular_authors":"","express_authors":"","created_at":"2024-08-14T10:38:09.042Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"expectation-of-a-random-variable","slug":"expectation-mean-and-variance-of-a-random-variable","header_name":"Expectation, Mean, and Variance of a Random Variable","type":"header-1","header_data":[],"_id":"66be8d6abe903046a22f2d8a"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"expectation-of-a-random-variable","slug":"expectations-of-jointly-distributed-random-variables","header_name":"Expectations of Jointly Distributed Random Variables","type":"header-1","header_data":[],"_id":"66be8d9cbe903046a22f2d9e"},{"sequence":3,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"expectation-of-a-random-variable","slug":"uncorrelated-and-independent-random-variables","header_name":"Uncorrelated and Independent Random Variables","type":"header-1","header_data":[],"_id":"66be8dc1be903046a22f2db2"},{"sequence":4,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"expectation-of-a-random-variable","slug":"expectations-of-sums-of-random-variables","header_name":"Expectations of Sums of Random Variables","type":"header-1","header_data":[],"_id":"66be8defbe903046a22f2dc6"},{"sequence":5,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"expectation-of-a-random-variable","slug":"the-law-of-large-numbers-and-the-central-limit-theorem","header_name":"The Law of Large Numbers and the Central Limit Theorem","type":"header-1","header_data":[],"_id":"66be8e0fbe903046a22f2dda"},{"sequence":6,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"expectation-of-a-random-variable","slug":"the-measurement-signal-to-noise-ratio-of-a-random-variable","header_name":"The Measurement Signal-To-Noise Ratio of a Random Variable","type":"header-1","header_data":[],"_id":"66be8e4abe903046a22f2dee"},{"sequence":7,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"expectation-of-a-random-variable","slug":"conditional-expectation--best-linear-prediction","header_name":"Conditional Expectation. Best Linear Prediction","type":"header-1","header_data":[],"_id":"66be8e67be903046a22f2e02"}]},{"_id":"66be8ecbbe903046a22f2e55","book_id":"66bd2adfbe903046a22f1be8","chapter_name":"Sums of Independent Random Variables","sequence":9,"chapter_content":"

Chapters 9 and 10 are much less elementary in character than the first eight chapters of this book. They constitute an introduction to the limit theorems of probability theory and to the role of characteristic functions in probability theory. These chapters seek to provide a careful and rigorous derivation of the law of large numbers and the central limit theorem.

In this chapter we treat the problem of finding the probability law of a random variable that arises as the sum of independent random variables. A major tool in this study is the characteristic function of a random variable, introduced in section 2. In section 3 it is shown that the probability law of a random variable can be determined from its characteristic function. Section 4 discusses some consequences of the basic result that the characteristic function of a sum of independent random variables is the product of the characteristic functions of the individual random variables. Section 5 gives the proofs of the inversion formulas stated in section 3.

","chapter_express_content":"","public_visibility":true,"book_slug":"probability-theory-and-its-applications","slug":"sums-of-independent-random-variables","regular_authors":"","express_authors":"","created_at":"2024-08-14T10:38:09.042Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"sums-of-independent-random-variables","slug":"the-problem-of-addition-of-independent-random-variables","header_name":"The Problem of Addition of Independent Random Variables","type":"header-1","header_data":[],"_id":"66be8eeebe903046a22f2e69"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"sums-of-independent-random-variables","slug":"the-characteristic-function-of-a-random-variable","header_name":"The Characteristic Function of a Random Variable","type":"header-1","header_data":[],"_id":"66be8f11be903046a22f2e7d"},{"sequence":3,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"sums-of-independent-random-variables","slug":"the-characteristic-function-of-a-random-variable-specifies-its-probability-law","header_name":"The Characteristic Function of a Random Variable Specifies Its Probability Law","type":"header-1","header_data":[],"_id":"66be8f2cbe903046a22f2e91"},{"sequence":4,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"sums-of-independent-random-variables","slug":"solution-of-the-problem-of-the-addition-of-independent-random-variables-by-the-method-of-characteristic-functions","header_name":"Solution of the Problem of the Addition of Independent Random Variables by the Method of Characteristic Functions","type":"header-1","header_data":[],"_id":"66be8f5dbe903046a22f2ea5"},{"sequence":5,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"sums-of-independent-random-variables","slug":"proofs-of-the-inversion-formulas-for-characteristic-functions","header_name":"Proofs of the Inversion Formulas for Characteristic Functions","type":"header-1","header_data":[],"_id":"66be8f7ebe903046a22f2eb9"}]},{"_id":"66be8f9ebe903046a22f2ecd","book_id":"66bd2adfbe903046a22f1be8","chapter_name":"Sequences of Random Variables","sequence":10,"chapter_content":"

The basic concepts of probability theory, such as the probability of a random event (or the mean of a random variable), have been given intuitive meanings as approximately representing certain averages computed from a large sample of independent observed values of the event (or of the random variable). In this chapter we treat the problem of giving an exact mathematical meaning to the word “approximately” as it is employed in the foregoing sentence. At the same time, our discussion leads to an answer to the question of what constitutes an approximate solution to the problem of finding the probability law of the sum of random variables. A basic role in this study is played by the notion of the convergence of a sequence of random variables .

","chapter_express_content":"","public_visibility":true,"book_slug":"probability-theory-and-its-applications","slug":"sequences-of-random-variables","regular_authors":"","express_authors":"","created_at":"2024-08-14T10:38:09.042Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"sequences-of-random-variables","slug":"modes-of-convergence-of-a-sequence-of-random-variables","header_name":"Modes of Convergence of a Sequence of Random Variables","type":"header-1","header_data":[],"_id":"66be8fbebe903046a22f2ee1"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"sequences-of-random-variables","slug":"the-law-of-large-numbers","header_name":"The Law of Large Numbers","type":"header-1","header_data":[],"_id":"66be8fdebe903046a22f2ef5"},{"sequence":3,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"sequences-of-random-variables","slug":"convergence-in-distribution-of-a-sequence-of-random-variables","header_name":"Convergence in Distribution of a Sequence of Random Variables","type":"header-1","header_data":[],"_id":"66be9000be903046a22f2f09"},{"sequence":4,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"sequences-of-random-variables","slug":"the-central-limit-theorem","header_name":"The Central Limit Theorem","type":"header-1","header_data":[],"_id":"66be9029be903046a22f2f1d"},{"sequence":5,"public_visibility":true,"header_type":"TEXT","book_slug":"probability-theory-and-its-applications","chapter_slug":"sequences-of-random-variables","slug":"proofs-of-theorems-concerning-convergence-in-distribution","header_name":"Proofs of Theorems Concerning Convergence in Distribution","type":"header-1","header_data":[],"_id":"66be9073be903046a22f2f31"}]},{"_id":"66be90e3be903046a22f2f45","book_id":"66bd2adfbe903046a22f1be8","chapter_name":"Tables","sequence":11,"chapter_content":"

In this chapter, we provide tables of normal distribution functions and binomial probabilities.

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Normal, Poisson, and Related Probability Laws

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