Chapter2_Probability_1

Chapter2_Probability_1 - 2. AXIOMS OF PROBABILITY 0.1....

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 Morgans 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 !...
View Full Document

Page1 / 12

Chapter2_Probability_1 - 2. AXIOMS OF PROBABILITY 0.1....

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online