If X and Y are symmetric random variables then sow that E(X/X+Y)= 1/2

Question

here E stands for expectation

Answer 1

Shivani:

 

Just one caviat to your question. ".....E(X/X+Y)= 1/2 provided the left side exists."

 

Rename X as Y and Y as X.  Then, by symmetry, E(X/X+Y) = E(Y/Y+X) = E(Y/X+Y)  CALL THIS (*).

 

Now,

 

E[(X+Y) / (X+Y)] ≡ 1.

 

But the left side is

 

E(X/X+Y) + E(Y/X+Y) =  2 E(X/X+Y) , by (*) above.

 

Then 2 E(X/X+Y) = 1.  Therefore, E(X/X+Y)= 1/2, as was to be shown.

 

Please post a comment about the proof above.

 

Dr, G,