ISEN 609 Lecture 4

ISEN 609 Lecture 4 - Joint(bivariate distribution function...

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Joint (bivariate) distribution function of X and Y Joint mass function and joint density function F ( x,y ) := P ( X x,Y y ) f ( x, y ) := x y F ( x, y ) p ( x, y ) = P ( X = x, Y = y )
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Remember how to get marginals for X and Y? F X ( x ) = lim y →∞ F ( x, y ) , F Y ( y ) = lim x →∞ F ( x, y ) f X ( x )= all y f ( x, y ) dy, f Y ( y all x f ( x, y ) dx p X ( x all y p ( x, y ) ,p Y ( y all x p ( x, y )
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Independence for random variables From the definition of independent events, we would need to check for each A and each B in the respective state spaces of X and Y . Fortunately, it suffices to check or equivalently for each x in the state space of X and each y in the state space of Y F ( x, y ) = F X ( x ) F Y ( y ) p ( x, y ) = p X ( x ) p Y ( y ) f ( x, y ) = f X ( x ) f Y ( y ) P ( X A,Y B )= P ( X A ) P ( Y B )
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4. Expectation (Expected Value) Expectation is an operator that returns a real number when evaluated on a function of a random variable (e.g., g ( X )). It can be computed from either P (the law) or F (in this course, we will always compute it from F ): where dF ( x ) = ± p ( x ) X discrete f ( x ) dx X continuous E [ g ( X )] := s S g ( X ( s )) P ( ds )= x g ( x ) dF ( x )
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This note was uploaded on 04/28/2011 for the course ISEN 609 taught by Professor Klutke during the Spring '08 term at Texas A&M.

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ISEN 609 Lecture 4 - Joint(bivariate distribution function...

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