This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full
Document
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).
 Summer '08
 WEBER

Click to edit the document details