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# l23 - Lecture 23 Conditional Densities The conditional...

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Lecture # 23 Conditional Densities: The conditional density for X given Y=y denoted by f X|y , ) y ( f ) y , x ( f ] y Y [ P ] y Y and x X [ P ] y Y and x X [ P Y XY = = = = = = =

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Def: Let (X,Y) be a two dimensional random variable with joint density f XY and marginal densities f X and f Y . Then 1. The conditional density for X given given Y=y, denoted by f X|y is given by: 0 ) y ( f , ) y ( f ) y , x ( f f Y Y XY y | X =
Note: The conditional densities satisfies all the requirements for a one dimensional pdf. Thus for a fixed y, we have g(x|y) 0 and - - - = = = = 1 ) x ( h ) x ( h dx ) y , x ( f ) y ( h 1 dx ) y ( h ) y , x ( f dx ) y | x ( g

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Example: Suppose that the two dimensional continuous random variable (X,Y) has joint pdf given by: elsewhere , 0 2 y 0 , 1 x 0 , 3 xy x ) y , x ( f 2 = + = The marginal density of X is:
x 3 2 x 2 dy ) 3 xy x ( dy ) y , x ( f ) x ( f 2 0 2 2 X + = + = = - The marginal density for Y is: 3 1 3 y dx ) 3 xy x ( ) y ( f 1 0 2 Y + = + =

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l23 - Lecture 23 Conditional Densities The conditional...

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