For random variables X , Y , I need to show that if X , Y are independent, th

tuehanhyd8ml

tuehanhyd8ml

Answered question

2022-05-15

For random variables X , Y, I need to show that if X , Y are independent, then E ( Y X ) = E ( Y ). To do so, it suffices to prove that for any B R measurable (Borel), we have that
X B E Y d P = X B Y d P
I know that X B E Y d P = E Y P ( X B ) - but how do I show that X B Y d P = E Y P ( X B )?

Answer & Explanation

aitantiskbx2v

aitantiskbx2v

Beginner2022-05-16Added 16 answers

Let A
being σ ( X ) −measurable. What has to be proved is
E [ Y 1 A ] = E [ E [ Y ] 1 A ] ,
but this is straightforward since
E [ Y 1 A ] = E [ Y ] E [ 1 A ] ,
since X and Y are independent.
othereyeshmt4l

othereyeshmt4l

Beginner2022-05-17Added 4 answers

If X and Y are independent, then f X Y ( x , y ) = f X ( x ) f Y ( y ) .
Since f Y | X = x ( x , y ) = f X Y ( x , y ) f X ( x ) = f X ( x ) f Y ( y ) f X ( x ) = f Y ( y )
E [ Y | X = x ] = R y f Y | X = x ( x , y ) d y = R y f Y ( y ) d y = E [ Y ] .
Now you can easily find E [ Y | X ] .

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