Summary 6 - Chapter 6 summary Jointly Distributed Random...

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Chapter 6 summary Jointly Distributed Random Variables Basic problem: Let X and Y be random variables. Consider the pair ( X,Y ). This is not a random variable because it isn’t a real-valued function (since its values are in R 2 , not R ). However, if D is any subset of R 2 , we’d like to know P (( X,Y ) D ), i.e. the probability that ( X,Y ) takes values in D . This is achieved in case X and Y are discrete by using the joint probability mass function p X,Y ( x,y ) and in case X and Y are continuous by using the joint probability density function f X,Y ( x,y ). These are each described next. Joint probability mass function: If X and Y are discrete random variables, the joint probability mass function is defined to be p X,Y ( x,y ) = P ( X = x and Y = y ) . For any subset D of the set of possible values of ( X,Y ), P (( X,Y ) D ) = X ( x,y ) D p X,Y ( x,y ) . The joint mass function has the property that it is 0 and x,y p X,Y ( x,y ) = 1 if the sum is taken over all possible pairs of values that ( X,Y ) can take. Joint probability density function: If X and Y are continuous random variables, the joint prob- ability density function is denoted by f X,Y ( x,y ). It has the property that it is 0 and for subsets D of R 2 for which we can define P (( X,Y ) D ) we have P (( X,Y ) D ) = Z D Z f X,Y ( x,y ) dA. The support of the joint probability density function is basically the set of ( x,y ) where f X,Y ( x,y ) > 0. More precisely, a point is in the support of f X,Y if for every disk centered at that point, the probability that ( X,Y ) is in that disk is strictly positive. In writing the above double integral as an iterated double integral, it is important to draw the region bounded by the intersection of
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This note was uploaded on 03/05/2011 for the course MATH 351 taught by Professor Moumen,f during the Spring '08 term at George Mason.

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Summary 6 - Chapter 6 summary Jointly Distributed Random...

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