If $U$ is a Uniform(0,1) random variable and $X\geq0$ is another random variable, then how do you determine $E(X\mid U(1-U))$?
Conditional Expectation on a function of a Random Variable
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probability-theory
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2If you don't know the JOINT distribution, or anything else about the relationship between $X$ and $U$, then you haven't really asked a question. – 2012-12-01
1 Answers
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- If $X$ is independent of $U$, then $\mathbb E(X\mid U(1-U))=\mathbb E(X)$.
- If $X=U$, then the distribution of $X$ conditionally on $U(1-U)=u$ is uniform on the two roots of the equation $x(1-x)=u$. Since these roots sum to $1$, $\mathbb E(X\mid U(1-U))=\frac12$.
- If $X=U(1-U)$, then $\mathbb E(X\mid U(1-U))=U(1-U)$.
These situations indicate that, as noted in the comments, some information about the joint distribution of $(U,X)$ is needed before the question even makes sense.
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0Unfortunately that was all the information given, other than $U$ being defined on $(\Omega, \mathcal{B}, P)$. But, thanks! – 2012-12-02