Lecture6

# Lecture6 - 1 TWO-DIMENSIONAL RANDOM VECTORS Lecture 6...

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Lecture 6 ORIE3500/5500 Summer2009 Chen Two-dimensional random vectors and discrete case 1 Two-dimensional random vectors Random vectors are also known as jointly distributed random variables. Most models require keeping track of more than one random variables at a time. We construct such a model by considering several random variables together X = ( X 1 ,X 2 ,...,X k ) . Let us begin with two dimensional random vectors. The probability distribution of a two dimensional random variable is char- acterized by its joint cumulative distribution function or joint cdf . It is de- ﬁned as F X,Y ( x,y ) = P [ X x,Y y ] , -∞ < x,y < . It is common to denote intersection of events involving random variables with ‘commas’. This means that in the deﬁnition above P [ X x,Y y ] is another way of writing P ([ X x ] [ Y y ]). F X,Y ( x,y ) is thus the probability of the region ‘south-west’ of the point ( x,y ). Properties of joint cdf Joint cdfs behave like ordinary cdf once we take into consideration the changes necessary since this is on a higher dimensional space. 1. For all ( x,y ), 0 F X,Y ( x,y ) 1. 2. It satisﬁes lim x →-∞ F X,Y ( x,y ) = 0 , y and lim y →-∞ F X,Y ( x,y ) = 0 , x. 3. Since the probability of the sample space is 1, we have lim x →∞ ,y →∞ F X,Y ( x,y ) = 1 . 1

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Lecture6 - 1 TWO-DIMENSIONAL RANDOM VECTORS Lecture 6...

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