1 properties of composite function for random

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Unformatted text preview: ite Function for Random Variable To satisfy that is a function of the RV, , it should satisfy three conditions (Papoulis and Pillai, 2002). Condition 1. Its domain must include the range of the RV, negative value, . If and have a remains undefined for such values. Thus, logarithm of a RV having negative values is not acceptable. Condition 2. It must be a Borel function. For every , the Set (satisfying ) must consist of the union and intersection of a countable number of intervals. This requirement is to satisfy that is an event. As defined in previous lecture, if the events are belong to a field or set, , then it is called as a Borel field or set if unions and intersections of these sets also belong to . Condition 3. The events must have zero probability 2. Determination of Density of Function Let us consider, . Now, for a specific value of , then is known, we have to determine . To find is solved. If there exists n real roots, +…+ +….where g’(x) is the derivative of g(x). Proof: Assume that the equation, Y g X has three roots as shown in Fig.1. As we know, . It suffices, therefore, to find the set of values of , such that and the probability that X is in this set. This set consists of three intervals , , dy . y...
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This note was uploaded on 03/18/2014 for the course CE 5730 taught by Professor Dr.rajibmaity during the Spring '13 term at Indian Institute of Technology, Kharagpur.

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