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36885 results in Cambridge Textbooks

Page 1508 of 1845

  • 8 - Detection, decisions, and hypothesis testing

  • Robert G. Gallager, Massachusetts Institute of Technology
  • Book:
    Stochastic Processes
    Published online:
    28 May 2018
    Print publication:
    12 December 2013, pp 375-416

  • Summary

    Detection, decision making, and hypothesis testing are different names for the same procedure. The word detection refers to the effort to decide whether some phenomenon is present or not in a given situation. For example, a radar system attempts to detect whether or not a target is present; a quality control system attempts to detect whether a unit is defective; a medical test detects whether a given disease is present. The meaning has been extended in the communication field to detect which one, among a finite set of mutually exclusive possible transmitted signals, has been transmitted. Decision making is, again, the process of choosing between a number of mutually exclusive alternatives. Hypothesis testing is the same, except the mutually exclusive alternatives are called hypotheses. We usually use the word hypotheses for these alternatives in what follows, since the word seems to conjure up the appropriate intuitive images.

    These problems will usually be modeled by a generic type of probability model. Each such model is characterized by a discrete random variable (rv) X called the hypothesis rv and a random vector (rv) Y called the observation rv. The observation might also be one-dimensional, i.e., an ordinary rv. The sample values of X are called hypotheses; it makes no difference what these hypotheses are called, so we usually number them, 0,1,…, M — 1.

Title match

  • 4 - Finite-state Markov chains

  • Robert G. Gallager, Massachusetts Institute of Technology
  • Book:
    Stochastic Processes
    Published online:
    28 May 2018
    Print publication:
    12 December 2013, pp 161-213

  • Summary

    Introduction

    The counting processes {N(t); t > 0} described in Section 2.1.1 have the property that N(t) changes at discrete instants of time, but is defined for all real t > 0. The Markov chains to be discussed in this chapter are stochastic processes defined only at integer values of time, n = 0,1,…. At each integer time n ≥ 0, there is an integer-valued random variable (rv) Xn , called the state at time n, and the process is the family of rv s {Xn; n ≥ 0}. We refer to these processes as integer-time processes. An integer-time process {Xn; n ≥ 0} can also be viewed as a process {X(t); t ≥ 0} defined for all real t by taking X(t) = Xn for n ≤ t ≤ n + 1, but since changes occur only at integer times, it is usually simpler to view the process only at those integer times.

    In general, for Markov chains, the set of possible values for each rv Xn is a countable set S. If S is countably infinite, it is usually taken to be S = {0,1,2,…}, whereas if S is finite, it is usually taken to be S = {1,…, M}. In this chapter (except for Theorems 4.2.8 and 4.2.9), we restrict attention to the case in which S is finite, i.e., processes whose sample functions are sequences of integers, each between 1 and M. There is no special significance to using integer labels for states, and no compelling reason to include 0 for the countably infinite case and not for the finite case. For the countably infinite case, the most common applications come from queueing theory, where the state often represents the number of waiting customers, which might be zero. For the finite case, we often use vectors and matrices, where positive integer labels simplify the notation. In some examples, it will be more convenient to use more illustrative labels for states.

  • Introduction

  • Herodotus
  • Edited by Simon Hornblower, All Souls College, Oxford
  • Book:
    Herodotus: Histories Book V
    Published online:
    16 October 2018
    Print publication:
    12 December 2013, pp 1-48

  • Summary

    (a) Book divisions

    Bks. 5 and 6 of Hdt., which contain the narratives of the Ionian revolt and the Marathon campaign, are central to the Histories. The present section will, after some remarks about book divisions, discuss the structure of bks. 5 and 6, both internally and in relation to the Histories as a whole.

    The first requirement is to think away the conventional book divisions altogether; such divisions seem generally to be a fourth-century innovation. There is no good reason to think that Herodotus divided his own work into nine books (unlike, say, Polybius, he does not use them himself to cross-refer or rather back-refer). The Herodotean division is probably Alexandrian i.e. Hellenistic, perhaps third or fourth century bc. We must distinguish two questions: who first says that Hdt.'s work was in nine books, and who first cited him by book number.

    The ‘chronographic’ source of Diodorus, which provided him with some good-quality historiographic and poetic dates, as well as king lists and dates of city-foundations and mergers (synoikisms), tells us that Hdt.’s work was in nine books.4 Diodorus himself wrote in the time of Julius Caesar or the early years of Augustus’ principate, but the chronographer worked in perhaps the second century bc, the time of Apollodoros the Chronicler (FGrHist 244). Apollodoros is, however, an unlikely candidate himself, as is Kastor of Rhodes (FGrHist 250), whose chronicle ended with Pompey’s triumph in 61 bc. It is better to leave the Diodoran chronographer without a name.

  • Suggestions for instructors and self study

  • Robert G. Gallager, Massachusetts Institute of Technology
  • Book:
    Stochastic Processes
    Published online:
    28 May 2018
    Print publication:
    12 December 2013, pp xix-xx

  • Summary

    The subject of stochastic processes contains many beautiful and surprising results at a relatively simple level. These results should be savored and contemplated rather than rushed. The urge to go too quickly, to sacrifice understanding for shallow bottom lines, and to cover all the most important topics should be resisted. This text covers all the material in two full term graduate subjects at MIT, plus many other topics added for enrichment, so it cannot be ‘covered’ in one term.

    My conviction is that if a student acquires a deep understanding of any, say, 20% of the material, then that student will be able to read and understand the rest with relative ease at a later time. Better still, a full appreciation of that 20% will make most students eager to learn more. In other words, instructors have a good deal of freedom, subject to a prerequisite structure, to choose topics of interest to them and their students to cover in a one term course.

    One of the two MIT courses leading to this text covers Chapters 1, 2, 4, 5, 6, 7, and 9, skipping many of the more detailed parts of the latter five chapters. The other course covers Chapters 1, 3, 8, and 10, again omitting many topics. The first course is largely discrete and the second largely continuous, and a different mix is probably more appropriate for a student taking only one subject.

  • Preface

  • Herodotus
  • Edited by Simon Hornblower, All Souls College, Oxford
  • Book:
    Herodotus: Histories Book V
    Published online:
    16 October 2018
    Print publication:
    12 December 2013, pp vii-viii

  • Summary

    This single-authored volume is planned as one of a pair with an edition of and commentary on bk. 6 in the same series, by the present author and Christopher Pelling in collaboration. Some sections of the Introduction to the present volume concern bk. 5 only; some concern both books; and some topics common to both books (Hdt. and Homer; Hdt.'s handling of Kleomenes and of Aigina) will be covered in the Introduction to bk. 6. Chronology will be covered in both volumes; for the distribution, see the Introduction, 3. For what is known or can be plausibly inferred about Herodotus’ life and travels, see S. West in Bowie 2007: 127–130.

    The groundwork for the commentary on book 5 was done as part of graduate (MA) teaching at University College London (UCL). In 2008–9, I taught books 5 and 7 jointly with Professor C. Carey, and in 2009–10, my last academic year at UCL, I taught books 5 and 6 on my own. I am grateful to Chris Carey for many insights and much shared enjoyment, and to all the students for their stimulating contributions.

Title match

  • 1 - What is phonology?

  • David Odden, Ohio State University
  • Book:
    Introducing Phonology
    Published online:
    28 May 2018
    Print publication:
    28 November 2013, pp 1-14

  • Summary

    PREVIEW

    KEY TERMS

  • sound

  • symbol

  • transcription

  • grammar

  • continuous nature of speech

    • This chapter introduces phonology, the study of the sound systems of language. Its key objective is to:

  • explain the difference between physical sound and “a sound” as a discrete element of language

  • highlight the tradeoff between accuracy and usefulness in representing sound

  • introduce the notion of “sound as cognitive symbol”

  • present the phonetic underpinnings of phonology

  • introduce the notion of phonological rule

    • Phonology is one of the core fields that compose the discipline of linguistics, which is the scientific study of language structure. One way to understand the subject matter of phonology is to contrast it with other fields within linguistics. A very brief explanation is that phonology is the study of sound structure in language, which is different from the study of sentence structure (syntax), word structure (morphology), or how languages change over time (historical linguistics). But this is insufficient. An important feature of the structure of a sentence is how it is pronounced – its sound structure. The pronunciation of a given word is also a fundamental part of the structure of the word. And certainly the principles of pronunciation in a language are subject to change over time. So phonology has a relationship to numerous domains of linguistics.

Page 1508 of 1845