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Conditional Expectation

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Conditional Distribution

If ~X and ~Y are random variables then the #~{conditional density} is

f ( ~y | ~x ) _ = _ f_{~Y|~X} ( ~y | ~x ) _ = _ f ( ~x , ~y ) / f ( ~x ) , _ _ providing _ f ( ~x ) != 0.

P ( ~a =< ~Y =< ~b ) _ = _ int{f_{~Y|~X} ( ~y | ~x ) ,~a ,~b, d~y}

f ( ~y | ~x ) f ( ~x ) _ = _ f ( ~x , ~y ) . _


If ~X and ~Y are random variables then

E ( ~Y | ~X = ~x ) _ = _ sum{~y ~f_{~Y|~X} ( ~y|~x ),~y, _ } _ = _ sum{~y P( ~Y = ~y |~X = ~x ),~y, _ }

 

Tower Property