Lecture 6,7 - Two Dimensional Random Vectors

Lecture 6,7 - Two Dimensional Random Vectors - 1...

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1 TWO-DIMENSIONAL RANDOM VECTORS Lecture 6 and 7 ORIE3500/5500 Summer2011 Li Two-dimensional random vectors 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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1 TWO-DIMENSIONAL RANDOM VECTORS 4. Joint cdf is non-decreasing in the sense that x 1 x 2 , y 1 y 2 , F X,Y ( x 1 , y 1 ) - F X,Y ( x 1 , y 2 ) - F X,Y ( x 2 , y 1 ) + F X,Y ( x 2 , y 2 ) 0 By the inclusion-exclusion principle this is same as saying that the probability of each rectangle is nonnegative. 5. It is right continuous. That is lim x n x,y n y F X,Y ( x n , y n ) = F X,Y ( x, y ) . One comment at this stage is that if a bivariate function satisﬁes these 5 conditions then it is the joint cdf of some bi-variate random vector. Note that if (
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Lecture 6,7 - Two Dimensional Random Vectors - 1...

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