Conditional Expectation

X is taken from [0,Pi]. What is E[X | sin X] ?

Solution:

Tests the understanding of conditional expectations.

First off, E[X|sin X] is a function, namely E[X|sin X] is E[X|sin X=y](y).

Figure out the sigma algebra generated by sin X. The interval [0,Pi] is our sample space, so sin X belongs to [0,1]. So sigma(sin X) is gonna be some Borel subset of the complete Borel set on [0, Pi].

The elements of this generated Borel subset are symmetric about Pi/2--for instance, take sin X = 0.5 --> X belongs to {Pi/6, 5*Pi/6}. This is evident, just graph sin X over [0, Pi]. So,

E[X|sin X=Y for any Y] = E[ X belongs to a set that is in the borel sets symmetric around Pi /2 ] = Pi/2.

which is independent of Y, the constant function Pi/2.