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Unformatted text preview: 2. AXIOMS OF PROBABILITY 0.1. Sample spaces and events. Consider a random experiment. The sample space of this experiment is the set of all possible outcomes, we will use S to senote a sample space. Each outcome in the sample space is called a sample point. Any subset E of the sample space is called an event. Example 0.1. Consider flipping a coin. The sample space is S = { H, T } , where H and T denote respectively the outcome that the coin lands up the heads and tails. Clearly, S has 4 events in total: ∅ , { H } , { T } and { H, T } = S . Example 0.2. Consider the experiment of measuring (in hours) the lifetime of a transistor, the sample space is S := { x : 0 ≤ x < ∞} . Then E := { x : 0 ≤ x ≤ 10 } denotes the event that the transistor does not last longer than 10 hours. We can define several operations on events as follows. (1) Union: E ∪ F , the union of two events E and F , is defined as the new event that consists of all outcomes that are either in E or in F . (2) Intersection. E ∩ F or EF , the intersection of two events E and F , is defined as the new event that consists of all outcomes belonging to both E and F . If E ∩ F = ∅ , we say they are disjoint or mutually exclusive. Given a sequence of evemts E 1 , E 2 , ··· , we can define the new events ∪ ∞ i =1 E i and ∩ ∞ i =1 E i as follows: ∩ ∞ i =1 E i consists of all outcomes that are in one of E i , and ∩ ∞ i =1 E i consists of all outcomes that are in all of E i . (3) Complement. E c , the complement of a given event E , is defined as the new event that consists of all outcomes that are not in E . For two events E and E , if all the outcomes in E are also in F , then we say E is contained in F , and write E ⊂ F . We say two events E and F are equal if E ⊂ F and F ⊂ E . Example 0.3. Consider tossing two dice, the sample space is S := { ( i, j ) : i, j = 1 , 2 , 3 , 4 , 5 , 6 } . Let E = { (1 , 50 , (1 , 6) , (2 , 4) , (3 , 3) , (5 , 2) } , F = { (2 , 5) , (1 , 6) , (3 , 3) , (5 , 1) } . Then E ∪ F = { (1 , 5) , (1 , 6) , (2 , 4) , (3 , 3) , (5 , 2) , (2 , 5) , (5 , 1) } , E ∩ F = { (1 , 6) , (3 , 3) } . Here are some properties of the above operations. • Commutative laws: E ∪ F = F ∪ E , E ∩ F = F ∩ E . 1 2 • Associate laws: ( E ∩ F ) ∩ G = E ∩ ( F ∩ G ) , ( E ∪ F ) ∪ G = E ∪ ( F ∪ G ) . • Distributive laws: ( E ∪ F ) ∩ G = ( E ∩ G ) ∪ ( F ∩ G ) , ( E ∩ F ) ∪ G = ( E ∪ G ) ∩ ( F ∪ G ) . • ( E c ) c = E . All these properties follow easily from the definition of the operations. Proposition 0.1. We have the following De Morgan’s laws. n [ i =1 E i ! c = n i =1 E c i ; (0.1) n i =1 E i ! c = n [ i =1 E c i . (0.2) Proof. We first prove (0.1). Note that x is an outcome in n [ i =1 E i !...
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 Spring '08
 Berg
 Probability, Probability theory, i=1 i=1

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