Lecture7 - 1 CONTINUOUS RANDOM VECTORS Lecture 7...

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Lecture 7 ORIE3500/5500 Summer2009 Chen Continuous random vectors 1 Continuous random vectors The definition of a continuous random vector is the analog of the definition of continuous random variable in the higher dimension. ( X,Y ) is a continuous random vector if it has a joint density, that is, there exists a bivariate function f X,Y ( x,y ) which satisfies P [( X,Y ) A ] = Z Z A f X,Y ( x,y ) dxdy, A R 2 . We can draw similarities immediately. The joint cdf and the joint pdf are related as F X,Y ( x,y ) = Z x -∞ Z y -∞ f X,Y ( s,t ) dsdt and 2 ∂x∂y F X,Y ( x,y ) = f X,Y ( x,y ) . If B = ( b 1 ,b 2 ] and C = ( c 1 ,c 2 ] are one dimensional sets and A is their cartesian product(the rectangle formed with sides B and C ), then P [( X,Y ) A ] = P [ X B,Y C ] = Z b 2 b 1 Z c 2 c 1 f X,Y ( x,y ) dxdy. Recall that for a continuous random variable X , we have ∂x F X ( x ) = f X ( x ) . Then P ( x X x + δ ) = F X ( x + δ ) - F X ( x ) f X ( x ) δ, which means f X ( x )is the probability per unit length. Now for two dimen- sional random vectors, P ( x X x + δ,y Y y + δ ) = Z y + δ y Z x + δ x f X,Y ( s,t ) dsdt
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Lecture7 - 1 CONTINUOUS RANDOM VECTORS Lecture 7...

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