Note for PROBABILITY AND RANDOM PROCESSES - PRP by Jack Reacher
- PROBABILITY AND RANDOM PROCESSES - PRP
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Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 01 Algebra of Sets – I Welcome to this course on probability and statistics. This is an introductory course or you can say first course on the probability and statistics, and it is quite useful for all branches of science and engineering. (Refer Slide Time: 00:36) So, to begin with, we introduce algebra of sets; this is required, because the modern theory of probability is based on the set theory. So, here we introduce certain elementary concepts of sets, which will be directly used in our definition and concepts of probability. So, to begin with, we start the discussion on what is set, classes and collections. So, we have, we start with a universal set; let us call it omega, and then, we also have the usual notation for the empty set as phi. We call a set of sets to be the class and a set of classes to be a collection.
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As an example, consider, suppose omega is the real line R and we consider the collection of intervals starting from a to b, where b is greater than a, for every real number a; then, this C a is a class of sets for each a. And then, if we collect all this classes C a into a set called script E, then this is a collection. For any two sets a and b which are subsets of omega, let us consider the relation a related to b; if a is a subset of B, then the relation R is reflective and transitive; it is symmetric, when omega is equal to phi. (Refer Slide Time: 02:35) We introduce some further conventions and notations. If the index set i is empty, we make the convention that union of E i is empty and also intersection of E i is the full set omega. The second of this convention looks to be rather surprising. However, it is motivated by the fact, that if we take more and more sets in an intersection, then it becomes smaller; for example, if I have 2 index sets I 1 and I 2, such that I 1 is a subset of I 2, then intersection of E i, where i belongs to I 1 contains intersection of E I, where i belongs to I 2, primarily because the more sets means the intersection becomes smaller. Hence, the smallest possible index set, that is, phi should lead to the largest intersection, that is omega. This convention is also consistent with De Morgan’s Laws; for example, if I write union of E i, i belongs to I compliment, is equal to intersection E i compliment i belonging to I. Let us take I to be phi, then the left hand side is equal to phi compliment is equal to x; this indicates that, we should take intersection of E i. When i is an a empty index set, then this should be equal to the full space omega
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(Refer Slide Time: 04:29) Let us introduce the concept of indicator function or characteristic function. The characteristic function of a set E is defined to be 1, if x belongs to E, and it is 0, if x does not belong to E. Alternatively, we can write the set E as the set of all those points, for which x chi E x is equal to 1; clearly, indicator function of the empty set is always 0 and the indicator function of the whole space is always 1. There are certain exercises here; for example, if a set a is a subset of B, then the indicator function of a is always less than or equal to indicator function of B. Conversely, if the indicator function of the set a is always less than or equal to the indicator function of B, then a is a subset of b. A simple proof of this is that, if we consider chi A x, then it is equal 1 for all x belonging to a. And since A is a subset of B, this implies that, chi B x is also 1 for these points. Now, for the points where chi A x is equal to 0 includes certain points, where chi B x may be 1 and for other points chi B x will also be 0. Therefore, chi A x is always less than or equal to chi B x. We have certain elementary properties of the characteristic functions; for example, characteristic function of an intersection is equal to the product of the characteristic functions of the two sets. It is also equal to the minimum of the indicator function values for those two sets. The proof can be simply obtained by definition; for example, chi of E intersection F x is equal to 1, if x belongs to both E and F, and 0 otherwise; that means, it
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is going to be equal to 1, only if chi E x and chi F x both are 1; and in every other case, it is going to be 0; so, that means, it is equal to the product. In a similar way, the minimum of chi E x and chi F x is going to be 1, only if both chi E and chi F x are 1; that means, chi E intersection F x is equal to 1. (Refer Slide Time: 07:03) Similarly, the indicator function relations for the unions, complimentations and differences are also there; for example, chi F E union F x is equal to chi E x plus chi F x minus chi F E intersection F x; it is also alternatively equal to the maximum of chi E x and chi F F of x; once again to look at the proof of this.
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