Chapter1 - Chapter 1 Elementary Probability Theory In this...

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Unformatted text preview: Chapter 1 Elementary Probability Theory In this introductory chapter a large number of concepts of theory of probability are intro- duced in an elementary way. This we do to achieve two objectives. For those readers who have some exposure to the subject this serves as a quick review. For others this chapter will prepare the background at an elementary level which will be helpful in understanding the concepts of probability and distributions developed in later chapters. 1.1 Modelling Random Phenomena In its earliest development the purpose of the theory of probability was limited mainly to the calculation of certain probabilities associated with the games of chance. Later with the introduction of a measure theoretic approach to probability it has developed into a fascinating mathematical theory whose ideas are now used in diverse scientific fields like physics, genetics, communication engineering, industrial engineering, sociology, psychol- ogy, economics, medicine, etc. The theory gains practical value and intuitive meaning in connection with real or conceptual experiments such as tossing a coin, throwing a die, drawing cards from a deck, observing thermal noise in electric circuits, observing phe- notypes of offspring when two species of plants are crossed, observing telephone traffic density, etc. or with phenomenon such as maintenance of quality of manufactured prod- ucts, studying effect of certain drugs, etc. In all these experiments and phenomena, the common property possessed by them is that each one of them may be considered a random experiment or phenomenon in the following sense. A random (or chance) phenomenon is an empirical phenomenon characterized by the property that its repeated observation under a given set of conditions or circumstances does not always lead to the same observed outcome as opposed to the deterministic phe- nomena . When a phenomenon is observed under a given set of conditions, we say that an experiment is performed or conducted. Thus, a deterministic experiment is charac- terized by the property that whenever the conditions of the experiment are replicated the outcome is the same so that there is a deterministic regularity . For example, if the temperature of water is raised to 100 C at normal atmospheric pressure, it will boil and 1 2 CHAPTER 1. ELEMENTARY PROBABILITY THEORY there will be no other outcome as long as this experiment is repeated under the same set of conditions. However, if the experiment consists of tossing a coin, for example, then our experience shows that sometimes a head and sometimes a tail comes up on repeated performance of this experiment so that there is no deterministic regularity. Nevertheless, one has the experience of noticing that the repetitions of this experiment result in different outcomes in such a way that there is statistical regularity , i.e., there exist numbers between 0 and 1 which represent the relative frequencies with which the different outcomes may be observed. Experiments possessing this characteristic property of showing statisticalbe observed....
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Chapter1 - Chapter 1 Elementary Probability Theory In this...

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