Introduction to Probability and Measure

verfasst von: K. R. Parthasarathy

Verlag: Hindustan Book Agency

Buchreihe : Texts and Readings in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Probability on Boolean Algbras

Abstract

In probability theory we look at all possible basic outcomes of a statistical experiment and assume that they constitute a set X, called the sample space. The points or elements of X are called elementary outcomes. We shall illustrate by a few examples.

K. R. Parthasarathy

Chapter 2. Extension of Measures

Abstract

In Section 1.11 we mentioned about the fruitfulness of introducing the idea of a collection of sets closed under countable set operations and introducing probability distributions on such a collection. To this end we introduce the following definition.

K. R. Parthasarathy

Chapter 4. Integration

Abstract

In the very first chapter we have seen the usefulness of integration of simple functions on a boolean space. In many problems of probability and statistics random variables which are not necessarily simple, do arise and it is necessary to define the ‘average value’ or ‘expectation’ of such quantities. This can be achieved by extending the notion of integral further. It is also worth noting that mechanical concepts like centre of mass, moment of inertia, work, etc., can be formulated precisely in terms of integrals. However, in the initial stages of its development, the theory of integrals received its first push from the hands of the French mathematician H. Lebesgue on account of many new problems that arose in analysing the convergence properties of Fourier series. In the present chapter we shall introduce the idea of integral with respect to a measure on any borel space and investigate its basic properties.

K. R. Parthasarathy

Chapter 5. Measures on Product Spaces

Abstract

We shall examine how measures can be constructed on product spaces out of what are called ‘transition measures’. The product measures turn out to be special cases of such a construction. Later we shall see how integration in the product space is reduced to successive integration over the marginal spaces. When the transition measures happen to be transition probability measures they acquire a practical significance of great value and form the foundations of the study of Markov processes.

K. R. Parthasarathy

Chapter 6. Hilbert Space and conditional Expectation

Abstract

In Section 4.7 we introduced the space L_p(µ) for p ≥ 1 and mentioned that they are Banach spaces. Among these the space L₂(µ) has a special role to play in our subject. In fact all statistical problems which centre around ‘extrapolation’ or ‘prediction’ are based on the fundamental properties of the space L₂(µ). To elaborate on this theme we shall develop the barest minimum of the theory of Banach and Hilbert spaces.

K. R. Parthasarathy

Chapter 7. Weak Convergence of Probability Measures

Abstract

Throughout this chapter we shall concern ourselves with the study of probability measures on separable metric spaces only. As usual, for any such metric space X we shall write B_X for the borel σ-algebra of subsets of X. We shall denote by C(X) the space of all bounded real valued continuous functions on X and M₀(X) the space of all probability measures on B_X.

K. R. Parthasarathy

Chapter 8. Invariant Measures on Groups

Abstract

Consider the space R^k with the Lebesgue measure L on the borel σ-algebra. We have seen earlier that

$$L\left( E \right) = L\left( {E + a} \right)\;for\;all\;a \in {R^k},\;E \in {B_{{R^k}}}$$

K. R. Parthasarathy

Backmatter

Titel: Introduction to Probability and Measure
verfasst von: K. R. Parthasarathy
Verlag: Hindustan Book Agency
Electronic ISBN: 978-93-86279-27-9
Print ISBN: 978-81-85931-55-5
DOI: https://doi.org/10.1007/978-93-86279-27-9