Conditional expectation of $X$ given $X+Y$












3














$X$ and $Y$ are two random independent variables, they have the same distribution and have finite expected value. Find $ mathbb{E} [X|X+Y] $



My attempt:



$$ mathbb{E}[X|X+Y] + mathbb{E}[Y|X+Y] = X + Y $$
Whats more $$ mathbb{E}[X|X+Y] = mathbb{E}[Y|X+Y] $$
because $X$ and $Y$ have the same distribution.



Hence:
$$ mathbb{E}[X|X+Y] = frac{1}{2}(X+Y) $$



Edit:



From comments below I conclude that this is a sufficient solution.










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  • 3




    If $X$ and $Y$ have the same distribution, this should hold by definition; they are the same function over $A$.
    – platty
    Nov 29 '18 at 22:21






  • 4




    Don't you have your answer at the point you say "Hence: $ mathbb{E}[X|X+Y] = frac{1}{2}(X+Y) $" ?
    – Henry
    Nov 29 '18 at 22:28
















3














$X$ and $Y$ are two random independent variables, they have the same distribution and have finite expected value. Find $ mathbb{E} [X|X+Y] $



My attempt:



$$ mathbb{E}[X|X+Y] + mathbb{E}[Y|X+Y] = X + Y $$
Whats more $$ mathbb{E}[X|X+Y] = mathbb{E}[Y|X+Y] $$
because $X$ and $Y$ have the same distribution.



Hence:
$$ mathbb{E}[X|X+Y] = frac{1}{2}(X+Y) $$



Edit:



From comments below I conclude that this is a sufficient solution.










share|cite|improve this question




















  • 3




    If $X$ and $Y$ have the same distribution, this should hold by definition; they are the same function over $A$.
    – platty
    Nov 29 '18 at 22:21






  • 4




    Don't you have your answer at the point you say "Hence: $ mathbb{E}[X|X+Y] = frac{1}{2}(X+Y) $" ?
    – Henry
    Nov 29 '18 at 22:28














3












3








3







$X$ and $Y$ are two random independent variables, they have the same distribution and have finite expected value. Find $ mathbb{E} [X|X+Y] $



My attempt:



$$ mathbb{E}[X|X+Y] + mathbb{E}[Y|X+Y] = X + Y $$
Whats more $$ mathbb{E}[X|X+Y] = mathbb{E}[Y|X+Y] $$
because $X$ and $Y$ have the same distribution.



Hence:
$$ mathbb{E}[X|X+Y] = frac{1}{2}(X+Y) $$



Edit:



From comments below I conclude that this is a sufficient solution.










share|cite|improve this question















$X$ and $Y$ are two random independent variables, they have the same distribution and have finite expected value. Find $ mathbb{E} [X|X+Y] $



My attempt:



$$ mathbb{E}[X|X+Y] + mathbb{E}[Y|X+Y] = X + Y $$
Whats more $$ mathbb{E}[X|X+Y] = mathbb{E}[Y|X+Y] $$
because $X$ and $Y$ have the same distribution.



Hence:
$$ mathbb{E}[X|X+Y] = frac{1}{2}(X+Y) $$



Edit:



From comments below I conclude that this is a sufficient solution.







probability






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 2 '18 at 14:06







Wywana

















asked Nov 29 '18 at 22:19









WywanaWywana

585




585








  • 3




    If $X$ and $Y$ have the same distribution, this should hold by definition; they are the same function over $A$.
    – platty
    Nov 29 '18 at 22:21






  • 4




    Don't you have your answer at the point you say "Hence: $ mathbb{E}[X|X+Y] = frac{1}{2}(X+Y) $" ?
    – Henry
    Nov 29 '18 at 22:28














  • 3




    If $X$ and $Y$ have the same distribution, this should hold by definition; they are the same function over $A$.
    – platty
    Nov 29 '18 at 22:21






  • 4




    Don't you have your answer at the point you say "Hence: $ mathbb{E}[X|X+Y] = frac{1}{2}(X+Y) $" ?
    – Henry
    Nov 29 '18 at 22:28








3




3




If $X$ and $Y$ have the same distribution, this should hold by definition; they are the same function over $A$.
– platty
Nov 29 '18 at 22:21




If $X$ and $Y$ have the same distribution, this should hold by definition; they are the same function over $A$.
– platty
Nov 29 '18 at 22:21




4




4




Don't you have your answer at the point you say "Hence: $ mathbb{E}[X|X+Y] = frac{1}{2}(X+Y) $" ?
– Henry
Nov 29 '18 at 22:28




Don't you have your answer at the point you say "Hence: $ mathbb{E}[X|X+Y] = frac{1}{2}(X+Y) $" ?
– Henry
Nov 29 '18 at 22:28










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