Lecture 5 - General Definitions of Random Variables

Lecture 5 - General Definitions of Random Variables - 1...

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Unformatted text preview: 1 RANDOM VARIABLES Lecture 5 ORIE3500/5500 Summer2011 Li 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 ∈ Ω : X ( s ) ∈ A } , which is a subset of Ω. 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....
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Lecture 5 - General Definitions of Random Variables - 1...

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