Conditinal Expectation Problem

[purplebird]

Given x,y and z are standard normal distributions with mean 0 and standard deviation 1. x,y and z are also statistically independent.

Find E{x|x+y+z=1}.

 

[Adeimantus]

symmetry of x, y, and z. For example, interchanging x and y in E[x| x+y+z=1] to get E[y|y+x+z = 1] doesn't change its value.

 

[purplebird]

symmetry of x, y, and z. For example, interchanging x and y in E[x| x+y+z=1] to get E[y|y+x+z = 1] doesn't change its value.

I am not able to understand what you are referring to. Could you please explain. Thanks.

 

[Adeimantus]

Sure thing... If I understand the problem correctly, it doesn't really matter what the exact distributions of x, y, and z are. It only matters that they are identically distributed. Also, note that the condition x+y+z = 1 is unchanged by permuting the letters x,y,z. This, together with their being identically distributed means that

E{x|x+y+z=1} = E{y|x+y+z=1} = E{z|x+y+z=1}


Also, think about what E{x+y+z|x+y+z=1} would be, and remember the additive property of the expected value.

edit: think simple. no hard integrals needed, which was the first thing that came to my mind when I read the problem.

 

[Adeimantus]

I left something out...it is also important that x,y,z are statistically independent. If, for example, x and y were statistically dependent, but x and z were not, then that would create an asymmetry and you could no longer conclude that the expected values of x, y, and z were equal.