Probability and Statistical Inference. Robert Bartoszynski

Probability and Statistical Inference

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Example 1.22

Take an experiment consisting of tossing a coin infinitely many times. The “natural” sample space images is the space of all infinite sequences images where images or 1 (or any other two distinct symbols representing heads and tails). The “simple” events here are of the form “heads on the images th toss,” that is, sets of all infinite sequences images with the images th coordinate images satisfying images . The events in the field generated by the simple events are of the form “heads on tosses images and tails on tosses images ,” with both images and images finite and the outcomes of all other tosses remaining unspecified.

An event that does not belong to this field, but does belong to the images ‐field generated by the simple events, is the event that “as the number of tosses increases, the frequency of heads approaches a limit.” Clearly, to determine whether or not this event occurs, it does not suffice to know any finite number of coordinates images .

To generalize this example, replace the outcome of the coin tosses by the result of some experiment repeated infinitely many times. This way the coordinate images carries more information than it does for the outcome of images th coin toss. The “simple” events are now of the form of sets of sequences images with images for images , while the images 's for images are unconstrained. Here images are events that occur at the first images times of observations. The “simple” events described above, of an obvious interest and importance both in applications and in building the theory, are called “cylinder” events. The smallest images ‐field containing all cylinder events comprises all events that may be of interest, including those that are obtained through limits of sequences of cylinder events.

Problems

1 1.4.1 Let be a countable partition of ; that is, for all , and . Let . Find .

2 1.4.2 Assume that John will live forever. He plays a certain game each day. Let be the event that he wins the game on the th day. (i) Let be the event that John will win every game starting on January 1, 2035. Label the following statements as true or false: (a) . (b) . (c) . (d) . (ii) Assume now that John starts playing on a Monday. Match the following events through with events through : John loses infinitely many games. When John loses on a Thursday, he wins on the following Sunday. John never wins on three consecutive days. John wins every Wednesday. John wins on infinitely many Wednesdays. John wins on a Wednesday. John never wins on a weekend. John wins infinitely many games and loses infinitely many games. If John wins on some day, he never loses on the next day.

3 1.4.3 Let be distinct subsets of . (i) Find the maximum number of sets (including and ) of the smallest field containing . (ii) Find the maximum number of sets in this field if . (iii) Answer (ii) if . (iv) Answer (ii) if . (v) Answer (i)–(iv) for a ‐field.

4 1.4.4 For let . Consider a sequence of numbers satisfying for all , and let . (i) Find and . (ii) Find conditions, expressed in terms of , under which exists, and find this limit. (iii) Define and . Answer questions (i) and (ii) for sequence .

5 1.4.5 Let be the set of all integers. For , let be the number of elements in the intersection . Let be the class of all sets for which the limitexists. Show that is not a field. [Hint: Let and { all odd integers between and and all even integers between and for . Show that both and are in but .]

6 1.4.6 Let . Show that the class of all finite unions of intervals of the form and , with possibly infinite or (intervals of the form etc.) forms a field.

Notes

1 1 Unless specifically stated, we will be assuming that all coins and/or dice tossed are fair (balanced).

2 2 Asterisks denote more advanced material, as explained in the Preface to the Second Edition.

3 3 In view of the fact proved earlier that all monotone sequences converge, this condition means that (a) if is an increasing sequence of sets in , then and (b) if is a decreasing sequence of sets in , then .

4 4 For various relations among classes of sets defined through closure properties under operations, for example, see Chow and Teicher (1997) and Chung (2001).

Chapter 2 Probability

2.1 Introduction

The concept of probability has been an object of debate among philosophers, logicians, mathematicians, statisticians, physicists, and psychologists for the last couple of centuries, and this debate is not likely to be over in the foreseeable future. As advocated by Bertrand Russell in his essay on skepticism, when experts disagree, the layman would do best by refraining from forming a strong opinion. Accordingly, we will not enter into the discussion about the nature of probability; rather, we will start from the issues and principles that are commonly agreed upon.

Probability is a number associated with an event that is intended to represent its “likelihood,” “chance of occurring,” “degree of certainty,” and so on. Probabilities can be obtained in several ways, the most common being (1) the frequency (or objective) interpretation, (2) the classical (sometimes called logical) interpretation, and (3) the subjective or personal interpretation of probability.

2.2 Probability as a Frequency

According to the common interpretation, probability is the “long‐run” relative frequency of an event. The

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