Probability and Statistical Inference. Robert Bartoszynski
are likely to remain indefinitely, our best and most reliable prediction tools.
To make decisions under uncertainty, one usually needs to collect some data. Data may come from past experiences and observations, or may result from some controlled processes, such as laboratory or field experiments. The data are then used to hypothesize about the laws (often called mechanisms) that govern the fragment of reality of interest. In our book, we are interested in laws expressed in probabilistic terms: They specify directly, or allow us to compute, the chances of some events to occur. Knowledge of these chances is, in most cases, the best one can get with regard to prediction and decisions.
Probability theory is a domain of pure mathematics and as such, it has its own conceptual structure. To enable a variety of applications (typically comprising of all areas of human endeavor, ranging from biological, medical, social and physical sciences, to engineering, humanities, business, etc.), such structure must be kept on an abstract level. An application of probability to the particular situation analyzed requires a number of initial steps in which the elements of the real situation are interpreted as abstract concepts of probability theory. Such interpretation is often referred to as building a probabilistic model of the situation at hand. How well this is done is crucial to the success of application.
One of the main concepts here is that of an experiment—a term used in a broad sense. It means any process that generates data which is influenced, at least in part, by chance.
1.2 Sample Space
In analyzing an experiment, one is primarily interested in its outcome—the concept that is not defined (i.e., a primitive concept) but has to be specified in every particular application. This specification may be done in different ways, with the only requirements being that (1) outcomes exclude one another and (2) they exhaust the set of all logical possibilities.
Consider an experiment consisting of two tosses of a regular die. An outcome is most naturally represented by a pair of numbers that turn up on the upper faces of the die so that they form a pair
Table 1.1 Outcomes on a pair of dice.
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| 1 | 2 | 3 | 4 | 5 | 6 | ||
| 1 | (1, 1) | (1, 2) | (1, 3) | (1, 4) | (1, 5) | (1, 6) | |
| 2 | (2, 1) | (2, 2) | (2, 3) | (2, 4) | (2, 5) | (2, 6) | |
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3 | (3, 1) | (3, 2) | (3, 3) | (3, 4) | (3, 5) | (3, 6) |
| 4 | (4, 1) | (4, 2) | (4, 3) | (4, 4) | (4, 5) | (4, 6) | |
| 5 | (5, 1) | (5, 2) | (5, 3) | (5, 4) | (5, 5) | (5, 6) | |
| 6 | (6, 1) | (6, 2) | (6, 3) | (6, 4) | (6, 5) | (6, 6) | |
In the case of an experiment of tossing a die three times, the outcomes will be triplets
Since the outcome of an experiment is not known in advance, it is important to determine the set of all possible outcomes. This set, called the sample space, forms the conceptual framework for all further considerations of probability.
Definition 1.2.1 The sample space, denoted by
Example 1.2
In Example 1.1, the sample space
Suppose that the only available information about the numbers, those that turn up on the upper faces of the die, is their sum. In such a case as outcomes, we take 11 possible values of the sum so that
For