2 Probability Why we need the probability theory in analyzing observed data? In the descrip- tive study of data considered in the last section, it is emphasized that the result cannot be generalized outside the observed data under consideration. Any ques- tion relating to the population from which the observed data were from cannot be answered within the descriptive statistics framework. In order to be able to do that we need the theoretical framework offered by probability theory. In ef- fect probability theory develops a mathematical model which provides the logical foundation of statistical inference procedures for analyzing observed dat In developing a mathematical model we must first identify the important fea- tures, relations and entities in the real world phenomena and then devise the concepts and choose the assumptions with which to project a generalized de- scription of there phenomena; an idealized pictures of these phenomena. The model as a consistent mathematical system has" a life of its own"and can be analyzed and studied without direct reference to real world phenomena.(Thinks of analyzing the population, we do not have to refer to the information in the sample By the 1920s there was a wealth of results and probability began to grow into a systematic body of knowledge. Although various people attempted a systemati zation of probability it was the work of the Russian mathematician Kolmogorov hich provided to be the cornerstone for a systematic approach to probability theory. Kolmogorov managed to relate the concept of the probability to that of a measure in integration theory and exploited to the full the analogies between set theory and the theory of functions on the one hand and the concept of random variable on the other. In a monumental monograph in 1933 he proposed an axiomatization of probability theory establishing it once and for all as part of mathematical proper. There is no doubt that this monograph provided to be the watershed for the later development of probability theory growing enormously in importance and applicability2 Probability Why we need the probability theory in analyzing observed data ? In the descrip￾tive study of data considered in the last section, it is emphasized that the results cannot be generalized outside the observed data under consideration. Any ques￾tion relating to the population from which the observed data were from cannot be answered within the descriptive statistics framework. In order to be able to do that we need the theoretical framework offered by probability theory. In ef￾fect probability theory develops a mathematical model which provides the logical foundation of statistical inference procedures for analyzing observed data. In developing a mathematical model we must first identify the important fea￾tures, relations and entities in the real world phenomena and then devise the concepts and choose the assumptions with which to project a generalized de￾scription of there phenomena; an idealized pictures of these phenomena. The model as a consistent mathematical system has ”a life of its own” and can be analyzed and studied without direct reference to real world phenomena. (Thinks of analyzing the population, we do not have to refer to the information in the sample.) By the 1920s there was a wealth of results and probability began to grow into a systematic body of knowledge. Although various people attempted a systemati￾zation of probability it was the work of the Russian mathematician Kolmogorov which provided to be the cornerstone for a systematic approach to probability theory. Kolmogorov managed to relate the concept of the probability to that of a measure in integration theory and exploited to the full the analogies between set theory and the theory of functions on the one hand and the concept of a random variable on the other. In a monumental monograph in 1933 he proposed an axiomatization of probability theory establishing it once and for all as part of mathematical proper. There is no doubt that this monograph provided to be the watershed for the later development of probability theory growing enormously in importance and applicability. 4