Probability and Statistical Inference. Robert Bartoszynski

Probability and Statistical Inference

href="#fb3_img_img_8c75d1e6-f7c2-513d-89d7-69028ebd1309.png" alt="images"/>HH, HT, TH}. The events “the results alternate” and “at least one head and one tail” imply one another, and hence are equal.

Definition 1.3.3 The set containing no elements is called the empty set and is denoted by images . The event corresponding to images is called a null (impossible) event.

Example 1.11 ^*²

The reader may wonder whether it is correct to use the definite article in the definition above and speak of “the empty set,” since it would appear that there may be many different empty sets. For instance, the set of all kings of the United States and the set of all real numbers images such that images are both empty, but one consists of people and the other of numbers, so they cannot be equal. This is not so, however, as is shown by the following formal argument (to appreciate this argument, one needs some training in logic). Suppose that images and images are two empty sets. To prove that they are equal, one needs to prove that images and images . Formally, the first inclusion is the implication: “if images belongs to images , then images belongs to images .” This implication is true, because its premise is false: there is no images that belongs to images . The same holds for the second implication, so images .

We now give the definitions of three principal operations on events: complementation, union, and intersection.

Definition 1.3.4 The set that contains all sample points that are not in the event images will be called the complement of images and denoted images , to be read also as “not images .”

Definition 1.3.5 The set that contains all sample points belonging either to images or to images (so possibly to both of them) is called the union of images and images and denoted images , to be read as “ images or images .”

Definition 1.3.6 The set that contains all sample points belonging to both images and images is called the intersection of images and images and denoted images .

An alternative notation for a complement is images or images , whereas in the case of an intersection, one often writes images instead of images .

The operations above have the following interpretations in terms of occurrences of events:

1 Event occurs if event does not occur.

2 Event occurs when either or or both events occur.

3 Event occurs when both and occur.

Example 1.12

Consider an experiment of tossing a coin three times, with the sample space consisting of outcomes described as HHH, HHT, and so on. Let images and images be the events “heads and tails alternate” and “heads on the last toss,” respectively. The event images occurs if either heads or tails occur at least twice in a row so that images , while images is “tails on the last toss,” hence, images . The union images is the event “either the results alternate or it is heads on the last toss,” meaning images . Observe that while images has two outcomes and images has four outcomes, their union has only five outcomes, since the outcome HTH appears in both events. This common part is the intersection images .

Some formulas can be simplified by introducing the operation of the difference of two events.

Definition 1.3.7 The difference, images of events images and images contains all sample

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Probability and Statistical Inference. Robert Bartoszynski