# Conditional expectations by conditioning on functions of random variables

by firemind   Last Updated August 01, 2020 10:19 AM - source

I have conjectured the following:

Let $f:\mathbb{R}\supseteq A \rightarrow B \subseteq \mathbb{R}$ be an injective function. Let $X$ be a random variable with support $A$ and $Y$ be some random variable that is not independent from $X$. Then, $$E[Y | X=x]=E[Y|f(X)=f(x)].$$

Is that correct? If it is correct, is there any "weaker" assumption (weaker than $f$ being injective) that would make this true?

Thanks.

Tags :

if $$Z=f(X)$$ and $$f$$ is injective function so $$\sigma(Z)=\sigma(X)$$

since $$Z=f(X)$$ so $$\sigma(Z) \subset \sigma(X)$$

and because $$f$$ is injective so $$X=f^{-1}(Z)=g(Z)$$ so $$\sigma(X) \subset \sigma(Z)$$

so $$\sigma(Z)=\sigma(X)$$ or $$\sigma(f(X))=\sigma(X)$$

now $$E(Y|X)=E(Y|\sigma(X))=E(Y|\sigma(f(X)))=E(Y|f(X))$$

Masoud
February 19, 2020 20:38 PM