Probability and Statistical Inference. Robert Bartoszynski
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Название: Probability and Statistical Inference

Автор: Robert Bartoszynski

Издательство: John Wiley & Sons Limited

Жанр: Математика

Серия:

isbn: 9781119243823

isbn:

СКАЧАТЬ by a pair of numbers that turn up on the upper faces of the die so that they form a pair images

, with

(see Table 1.1).

Table 1.1 Outcomes on a pair of dice.


	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)
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 images , with images , images , and images being integers between 1 and 6.

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 images , is the set of all outcomes of an experiment. The elements of the sample space are called elementary outcomes, or sample points.

Example 1.2

In Example 1.1, the sample space images has images sample points in the case of two tosses, and images points in the case of three tosses of a die. The first statement can be verified by a direct counting of the elements of the sample space. Similar verification of the second claim, although possible in principle, would be cumbersome. In Chapter 3, we will introduce some methods of determining the sizes of sets without actually counting sample points.

Example 1.3

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 instance, all outcomes on the diagonal of Table 1.1—(6, 1), (5, 2), (4, 3), (3, 4), (2, 5), and (1, 6)—are represented by the same value 7.

Example 1.4

If we are interested in the number of accidents that occur at a given intersection within a month, the sample space might be taken as the set images consisting of all nonnegative integers. Realistically, there is a practical limit, say images , of the monthly numbers of accidents at this particular intersection. Although one may think that it is simpler to take the sample space images it turns out that it is often much simpler to take the infinite sample space if the “practical bound” is not very precise.

Since outcomes can be specified in various ways (as illustrated by Examples 1.1 and 1.3), it follows that the same experiment can be described in terms of different sample spaces images . The choice of a sample space depends on the goal of description. Moreover, certain sample spaces for the same experiment lead to easier and simpler analysis. The choice of a “better” sample space requires some skill, which is usually gained through experience. The following two examples illustrate this point.

Example 1.5

Let the experiment consist of recording the lifetime of a piece of equipment, say a light bulb. An outcome here is the time until the bulb burns out. An outcome typically will be represented by a number images СКАЧАТЬ

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