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ecen303_lec8

# ecen303_lec8 - ECEN 303 Random Signals and Systems Lecture...

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ECEN 303: Random Signals and Systems Lecture 8: Jointly Distributed Random Variables 1 Joint Distribution Functions Thus far, we have only concerned ourselves with probability distributions for single random variables. However, we are often interested in probability statements concerning two or more random variables. In order to deal with such probabilities, we deﬁne, for any two random variables X and Y , the joint CDF of X and Y by F ( x,y ) = Pr( X x,Y y ) The CDF of X and Y can be obtained from the joint CDF of X and Y as follows: F X ( x ) = Pr( X x ) = Pr( X x,Y < ) = F ( x, ) and F Y ( y ) = Pr( Y y ) = Pr( X < ,X x ) = F ( ,y ) The distribution functions F X ( · ) and F Y ( · ) are usually referred to as the marginal CDF of X and Y , respectively. All joint probability statements about X and Y can, in theory, be answered in terms of their joint CDF. For example, for any x 1 < x 2 and y 1 < y 2 , Pr( x 1 < X x 2 ,y 1 < Y y 2 ) = Pr( X x 2 ,Y y 2 ) - Pr( X x 2 ,Y y 1 ) - Pr( X x 1 ,Y y 2 ) + Pr( X x 1 ,Y y 1 ) = F ( x 2 ,y 2 ) - F ( x 2 ,y 1 ) - F ( x 1 ,y 2 ) + F ( x 1 ,y 1 ) In case when both X and Y are discrete random variables, it is convenient to deﬁne the joint PMF of X and Y by p ( x,y ) = Pr( X = x,Y = y ) 1

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For any set A of pairs of real numbers, Pr(( X,Y ) A ) = X ( x,y ) A p ( x,y ) In particular, the (marginal) PMF of X can be obtained from p ( x,y ) as p X ( x ) = Pr( X = x ) = X y p ( x,y ) Similarly, p Y ( y ) = Pr( Y = y ) = X x p ( x,y ) We say that X and Y are jointly continuous if there exists a function f ( x,y ), deﬁned for all real x and y , having the property that for every set A of pairs of real numbers Pr(( X,Y ) A ) = Z Z A f ( x,y ) dxdy The function f ( x,y ) is called the
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ecen303_lec8 - ECEN 303 Random Signals and Systems Lecture...

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