Designed for students studying mathematical statistics and probability after completing a course in calculus and real variables, this text deals with basic notions of probability spaces, random variables, distribution functions and generating functions, as well as joint distributions and the convergence properties of sequences of random variables. Includes worked examples and over 250 exercises with solutions.
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Yes, you can access Probability by C. R. Heathcote in PDF and/or ePUB format, as well as other popular books in Mathematics & Probability & Statistics. We have over one million books available in our catalogue for you to explore.
Probability theory is concerned with the mathematical analysis of quantities derived from observations of phenomena whose occurrence involves a chance element. The result of tossing a ‘fair’ coin cannot be predicted with certainty and any mathematical theory of probability must give meaning to and answer questions such as ‘what is the probability of n heads in N tosses of such a coin?’ Coin tossing is a simple example of the class of phenomena with which the theory is concerned.
For a mathematical theory the essential ingredients are that an act or experiment is performed (e.g. tossing a coin) all possible outcomes of which can at least in principle be specified and are observable (heads or tails), and that a rule is given whereby for each possible outcome or set of outcomes there can be defined in a consistent way a number called the probability of the outcome or set of outcomes in question.
Given a chance phenomenon the set of possible outcomes is called the sample space and will be denoted by Ω. The individual outcomes, generically denoted by ω, comprise the elements of the sample space, thus Ω = {ω}. Our first concern will be to set up the requisite formalism for an axiomatic definition of probability. This is done in § 1.1 commencing with the simple case of a sample space containing only finitely many elements.
1.1 PROBABILITY SPACES
Suppose we are given a sample space Ω consisting of the m points ω1, ω2, . . ., ωm;
Ω = {ω1, ω2, . . . , ωm}.
Example 1 One toss of a coin. The only possible outcomes are heads or tails. Let ω1 = H denote heads and ω2 = T denote tails. Then the sample space Ω contains two elements, Ω = {H, T}.
Example 2 Roll of a die. Supposing the six sides of the die are numbered 1 to 6, Ω consists of the first six positive integers.
Example 3 Roll of two dice. An outcome ωj is a vector (a, b) in which a exhibits the number shown by the first die and b that shown by the second. Ω contains 62 = 36 ele...
Table of contents
Title Page
Copyright Page
Table of Contents
PREFACE
Chapter 1 - PROBABILITY SPACES AND RANDOM VARIABLES
