2Basic_Concepts

2Basic_Concepts - Chapter 2 Basic Concepts of Probability...

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Unformatted text preview: Chapter 2 Basic Concepts of Probability The theory of probability provides the mathematical tools necessary to study and analyze uncertain phenomena that occur in nature. It establishes a formal framework to understand and predict the outcome of a random experiment. It can also be used to model complex systems and characterize stochastic pro- cesses. This is instrumental in designing efficient solutions to many engineering problems. Two components define a probabilistic model, a sample space and a probability law. 2.1 Sample Spaces and Events In the context of probability, an experiment is a random occurrence that pro- duces one of several outcomes . The set of all possible outcomes is called the sample space of the experiment, and it is denoted by Ω. An admissible subset of the sample space is called an event . Example 1. The rolling of a die forms a common experiment. A sample space Ω corresponding to this experiment is given by the six faces of a die. The set of prime numbers less than or equal to six, namely { 2 , 3 , 5 } , is one of many possible events. The actual number observed after rolling the die is the outcome of the experiment. There is essentially no restriction on what constitutes an experiment. The flipping of a coin, the flipping of n coins, and the tossing of an infinite sequence of coins are all random experiments. Also, two similar experiments may have 8 2.1. SAMPLE SPACES AND EVENTS 9 1 2 3 4 5 6 7 event outcome Figure 2.1: A sample space contains all the possible outcomes; an admissible subset of the sample space is called an event. Figure 2.2: A possible sample space for the rolling of a die is Ω = { 1 , 2 , . . . , 6 } , and the subset { 2 , 3 , 5 } forms a specific event. different sample spaces. The sample space Ω for observing the number of heads in n tosses of a coin is { , 1 , . . . , n } ; however, when describing the complete history of the n coin tosses, the sample space becomes much larger with 2 n distinct sequences of heads and tales. Ultimately, the choice of a particular sample space depends on the properties one wishes to analyze. Yet some rules must be followed in selecting a sample space. 1. The elements of a sample space should be distinct and mutually exclusive . This insures that the outcome of an experiment is unique. 2. A sample space must be collectively exhaustive . That is, every possible outcome of the experiment must be accounted for in the sample space. In general, a sample space should be precise enough to distinguish between all outcomes of interest, while avoiding frivolous details. 10 CHAPTER 2. BASIC CONCEPTS Example 2. Consider the space composed of the odd integers located between one and six, the even integers contained between one and six, and the prime numbers less than or equal to six. This collection cannot be a sample space for the rolling of a die because its elements are not mutually exclusive. In particular, the numbers three and five are both odd and prime, while the number two is prime and even. This violates the uniqueness criterion.two is prime and even....
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This note was uploaded on 03/30/2010 for the course ECEN 303 taught by Professor Chamberlain during the Fall '07 term at Texas A&M.

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2Basic_Concepts - Chapter 2 Basic Concepts of Probability...

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