Lecture5 - 1 RANDOM VARIABLES Lecture 5 ORIE3500/5500...

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Unformatted text preview: 1 RANDOM VARIABLES Lecture 5 ORIE3500/5500 Summer2009 Chen General Definitions of Random Variables 1 Random Variables A random variable is a real valued function defined on the sample space, that is, a random variable assigns a value for every element in the sample space. One typically uses capital letters and most often X,Y,Z etc. to denote random variables. Often these will have subscripts to describe a lot of random variables. Recall that we defined events as subsets of sample space. For a set A of the real numbers [ X A ] is the short form of the event { s : X ( s ) A } , which is a subset of . Definition. The cumulative distribution function(cdf) of a random variable X is defined as the function F X ( x ) = P [ X x ] ,- < x < . Often the subscript X in the cdf F X will not be used, particularly, when the random variable in question is clear from the context. The cdf fully defines the probability distribution of the random variable X . This means that for every relevant set A of the real numbers we can find P [ X A ] once the cdf is given. For example if A = ( a,b ], then P [ X A ] = P [ a < X b ] = P [( X b ) ( X a ) c ] = F X ( b )- F X ( a ) . Properties of CDF 1. F ( x ) is between 0 and 1. 2. F ( x ) is nondecreasing, i.e, if x y , then F ( x ) F ( y ). 3. lim x - F ( x ) = 0 , lim x F ( x ) = 1 ....
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This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell University (Engineering School).

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Lecture5 - 1 RANDOM VARIABLES Lecture 5 ORIE3500/5500...

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