Let X and Y be two dependent random variables with unknown dependency structure.
Is it possible to derive the distribution of E[X|Y] with minimal information? what information is necessary for this task? Should the distribution be similar to the distribution of X?
Is it naive to assume that it should at least be bell-curved? If samples of X|Y are given, then S_n converges to E[X|Y]. The central limit theorem for conditional expectation ensures the convergence to normal distribution. But assuming that distributions of S_n and E[X|Y] is similar, is not necessarily the same thing.
Do you know any papers, which try to handle distributions of X, Y and E[X|Y]?
I expect that the use of minimum cross-entropy principle (minimum Kullback-Leibler divergence) is the appropriate methodology for solving this task. See
@ARTICLE{ShoJon:80,
author = {J. Shore and R. Johnson},
title = {Axiomatic derivation of the principle of maximum entropy \& the principle of minimum cross-entropy},
journal = {IEEE\,Tran.\,on\,Inf.\,Th.},
year = {1980},
volume = {26},
number = {1},
pages = {26--37}
}
or
@incollection{KarGuy:12,
author={M. K{\'{a}}rn{\'{y}} and T.V. Guy},
title={On Support of Imperfect {B}ayesian Participants},
editor={T.V. Guy and et al},
booktitle= {Decision Making with Imperfect Decision Makers},
publisher={Springer},
address= {Berlin},
year={2012},
volume={28},
note={{I}ntelligent {S}ystems {R}eference {L}ibrary}}
Dear Dr. Christopher Paulus Imanto
Greating!
You can read the following links that may be useful to answer
https://www.khanacademy.org/math/ap-statistics/random-variables-ap/discrete-random-variables/v/discrete-probability-distribution
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https://www.stat.pitt.edu/stoffer/tsa4/intro_prob.pdf
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https://en.wikipedia.org/wiki/Normal_distribution
......
https://en.wikipedia.org/wiki/Probability_distribution
............
http://www.stat.ucla.edu/~nchristo/statistics100B/stat100b_continuous_dist.pdf
...........
http://www.stat.ucla.edu/~rosario/classes/081/100a-2a/100aHW6Soln.pdf
............
http://booksee.org/book/1144581
.............
http://booksee.org/book/448519
.........
http://booksee.org/book/547987
.............
https://math.mit.edu/~dav/05.dir/class5.5-prep.pdf
With best wishes.
Dear Mr. A. M. Abdallah ,
thanks for your suggestion. I highly doubt that the question has an elementary answer as you probably suggested. However, I'm interested in your opinion, whether you think I may miss something very simple here.
Assuming the dependency between X and Y is known, then this can be solved theoretically by solving an integral. If not (as stated in the question) the density of X|Y remains unknown. There is no guarantee that both densities (of X|Y and X) are similar.
However, thanks to you A. M. Abdallah
I was so focused on the question having a complex answer and disregard any simple solution. An easy counter-example is sufficient to answer that the distribution of E[X|Y] is neither similar to X nor has it to be bell-curved.
If X|Y ~ N(Y, \sigma), then E[X|Y] = Y. Y can be chosen for any non-bell-curved distribution to dismiss the hypothesis of it being a bell-curved. From theoretical point-of-view, the question might not have an answer (as in no specific property without further assumptions).
Christopher Paulus Imanto
Congrats! You just answered your question. The example has an extention to the following construction (and even to some general constructions):
- take any family of continuous densities on R with mean 0, dependent continuously on y as a parameter: f(x|y), x,y \in R;
\int_R x f(x|y) dx =0
- assume any continuous density function g(y)
- define h(x,y):= f(x-y)|y) g(y) x,y \in R
Properties of h:
- h is a probability density function on R^2
- any pair of random variables (X,Y) with this distribution obeys E(X|Y) = Y
Corollary:
The pd of E(X|Y) possesses density g.
Remark:
the assumptions on continuity can be weakened by suitable measurability conditions making the question a starred problem (for advanced students in probability and/or analysis:).
Remark:
Q: Can you recover the properties of function h of two variables from the properties of g?
A: Yes, but nothing more than that:
\int_R x h(x,y) dx / \int_R h(x,y) dx is for almost all y equal y.
Best regards, Joachim
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