Probability and Statistical Inference. Robert Bartoszynski

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СКАЧАТЬ 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 , then the events and are called disjoint, or mutually exclusive.

Example 1.14

      Based on Example 1.12, we know that the following two events are disjoint: “more heads than tails” and the intersection “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, 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)

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

(1.2)

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

      Example 1.15

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

      A perceptive reader may note that the unions and intersections 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 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)

      In particular,

which in view of (1.3) means that .

      Commutativity:

      Associativity:

      Distributivity:

      De Morgan's Laws:

(1.4)

It is often helpful to use Venn diagrams for studying relations between composite events in the same sample space. The sample space СКАЧАТЬ