Lecture7 - A Vector Random Variable Experiments with more...

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•1 A Vector Random Variable Lecture 7 A Vector Random Variable Experiments with more than one measurement Example: The ftp experiment 2192 bytes received in 0.11 seconds (20 kbytes/s) 26573 bytes received in 0.46 seconds (56 kbytes/s) Defining X=( size, time, speed), we can write X 1 = (2192, 0.11, 20) X 2 =(26573, 0.46, 56) It is clear that the vector X is random, since the values of components cannot be predicted for sure. Motivation Example: GRE scores(in percents) Let X=( verbal, analytical, quantitative). Then we can get the following random vectors: X 1 = (90, 75, 80) X 2 =(95, 94, 80) Note! The components of a random variable may have some relationship (dependence) with each other or not. It is only important they all be of a probabilistic nature. Product Events Sets Sets of the form play a fundamental role in 1-d cases Used to construct Borel sets on R used to define random variables, cdf functions etc. Similar approach can be used in higher dimensional spaces For simplicity concentrate on 2- dimensional case Everything may be generalized for spaces of dimensions 2, 3…. ] , ( ] , ( y x −∞ × −∞ ] , ( x −∞
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•2 Product Events Definition 1 Let us consider a collection of sets Then the smallest σ -algebra that contains all sets from A is called a Borel σ -algebra and denoted as . Mathematically, if , then it is called a Borel set . , , , ]} , ( ] , {( } { R y x R y x y x y x A −∞ × −∞ = ) ( ) ( A R B σ = 2 ) ( 2 R B B 2 on R Other Product Events Representation of other subsets of Note that
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Lecture7 - A Vector Random Variable Experiments with more...

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