I have never seen this notion, but i know E(X|Y). In this link i encounter the notion as X|Y and he said that X|Y has mean E(X|Y). He also give a property that if X and Y independent then X|Y=X. I only know that if X and Y indepentdent then E(X|Y)=EX and if X in \sigma(Y) (algebra generated by Y) then E(X|Y)=X.
https://stanford.edu/class/ee363/lectures/kf.pdf
Infact, the notation X|Y made me very confused.! What is the exactly definition of X|Y?
Thank so much!
Thanks so much your answer! But my question is that what does it means the notion X|Y without E or P, i understand E(X|Y), P(X|Y) but never seen only X|Y
For sake of simplicity, suppose that the conditional density function h(x|y):=f_{Y,X}(y,x)/f_Y(y) exists. Now I "define" X|Y as follow: If we "observe" Y (or in other word, Y is known, i.e. Y=y which is a generic constant), then X|Y is a random variable with density function g(x)=h(x|Y).
The exact definition needs a bit abstract algebra, but is quite straightforward if you already get the intuition behind the concept of conditional distribution.
A well-known example is the case of classical linear regression: If we have a model X=aY+b +error, then X|Y is a normal random variable with mean aY+b and standard deviation equal to that of the error, so long as Y is observed.
Thank so much for all answers!
@Peter Breuer : Sorry because that my english is not good!
I mean that i have never seen the definition of variable X|Y before reading the paper in the link attachment.
@Tuan Q.Tran: Your explanation is very clear for me! X|Y is well defined! :D
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