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handout_week5_a - Stochastic Signals and Systems Multiple...

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Stochastic Signals and Systems Multiple Random Variables Virginia Tech Fall 2008 Multiple Random Variables We now develop techniques for calculating the probabilities of events that involve the joint behavior of two or more rvs. We are also interested in determining when a set of rvs are independent, as well as in quantifying their degree of correlation ” when they are not independent. Example: Given two random variables X and Y we wish to determine their joint statistics, that is, the probability that the point ( X , Y ) is in a specified region D in the xy plane. In particular, the probability of the event { X x } ∩ { Y y } = { X x , Y y } cannot be expressed in terms of F X ( x ) and F Y ( y ) .
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Joint Cumulative Distribution Function For any pair of numbers x and y , the event ( X x , Y y ) is represented by a quadrant of the XY-plane having its vertex at the point ( x , y ) and opening to the lower left, as shown below. The probability of this event is denoted by F X , Y ( x , y ) = P [ X x , Y y ] and is called the joint cumulative distribution function of X and Y
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