Probability and Statistical Inference. Robert Bartoszynski

Probability and Statistical Inference

rel="nofollow" href="#fb3_img_img_d5c715c1-a121-5574-8365-5908b7260603.png" alt="images"/> but not to images

The symmetric difference, images , contains sample points that belong to images or to images , but not to both of them:

Example 1.13

In Example 1.12, the difference images is described as “at least two identical outcomes in a row and tails on the last toss,” which means the event {HHT, HTT, TTT}.

Next, we introduce the following important concept:

Definition 1.3.8 If images , then the events images and images are called disjoint, or mutually exclusive.

Example 1.14

Based on Example 1.12, we know that the following two events are disjoint: images “more heads than tails” and the intersection images “the results alternate, ending with tails.”

Example 1.14 shows that to determine whether or not events are disjoint, it is not necessary to list the outcomes in both events and check whether there exist common outcomes. Apart from the fact that such listing is not feasible when sample spaces are large, it is often simpler to employ logical reasoning. In the case above, if the results alternate and end with tails, then the outcome must be THT. Since there are more tails than heads, images does not occur.

The definitions of union and intersection can be extended to the case of a finite and even infinite number of events (to be discussed in the Section 1.4). Thus,

(1.1) equation

is the event that contains the sample points belonging to images or images or …or images . Consequently, (1.1) is the event “at least one images occurs.” Similarly,

(1.2) equation

is the event that contains the sample points belonging to images and images and …and images . Consequently, the event (1.2) is “all images 's occur.”

Example 1.15

Suppose that images shots are fired at a target, and let images , images denote the event “the target is hit on the images th shot.” Then, the union images is the event “the target is hit” (at least once). Its complement images is the event “the target is missed” (on every shot), which is the same as the intersection images .

A perceptive reader may note that the unions images and intersections images do not require an extension of the definition of union and intersection for two events. Indeed, we could consider unions such as

where the union of only two events is formed in each set of parentheses. The property of associativity (below) shows that parentheses can be omitted so that the expression images is unambiguous. The same argument applies to intersections.

The operations on events defined in this section obey some laws. The most important ones are listed below.

Idempotence:

Double complementation:

Absorption:

(1.3) equation

In particular,

which in view of (1.3) means that images .

Commutativity:

Скачать книгу