Lecture3 - Random Variables Lecture 3 Random Variables and...

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1 Lecture 3 Random Variables and their Statistics Random Variables ± Recall we wanted an algebraic structure to combine various possible events from the sample space σ− field . ± To use the computational engine of calculus and set up a deductive ( conclude and predict) framework We map events from sample space to \ Definition of sets on ± How should sets and subsets be represented ² To accommodate unions and intersections? ² A function of one argument can be defined? ³ Representation of subsets of by : Ex1: Suppose we want to represent a set with sets. \ \ (, ] x −∞ (,] ab ] x −∞ Continued ± Ex2: ± Ex3: ] ] ba = −∞ − −∞ ] ] c =−∞ −∞ 11 {} l im{ ( , )} n aa a nn →∞ =− + [,] (,] {} a = {( , ] ( , ] {lim ( , ]} c n a a →∞ + ∩∪
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2 Summary ± N-B: For all practical purposes, then any result on the real line can be represented by sets ± These sets are referred to as Borel sets ± The collection of such sets is a Borel σ− field ( or σ− algebra) (, ] x −∞ Concept of a Random variable ± Suppose we conduct an experiment of drawing balls of a bag “full” of multicolor balls. To each event or set of events will correspond a Real variable or a set of real values in R (i.e. a segment) ¾ Through this mapping, we can measure the probability of the various events via the probability of subsets/ sets of R x \ Formal Definition ± A “ random variable” is a mapping from a sample space onto the real line , such that sets in B( ) are mapped onto sets in : or equivalently for every outcome X Ω \
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This note was uploaded on 09/24/2009 for the course ECE 514 taught by Professor Krim during the Fall '08 term at N.C. State.

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Lecture3 - Random Variables Lecture 3 Random Variables and...

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