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In my probability class I was tackled with this seemingly weird question involving conditional expectation:

Let X,Y be two random variables (it is not mentioned whether or not they are discrete or continuous) and we are asked the following:

For all constants $ \beta $ we have $ E[X | Y = \beta] = E[X] $

- Version of Conditional Expectation
- Computing the expectation of conditional variance in 2 ways
- $X = E(Y | \sigma(X)) $ and $Y = E(X | \sigma(Y))$
- Is conditional expectation with respect to two sigma algebra exchangeable?
- sign of the conditional expectation
- If $E(X\mid Y)=Z$, $E(Y\mid Z)=X$ and $E(Z\mid X)=Y$, then $X=Y=Z$ almost surely
$ cov(X,Y) = E[XY]-E[X]E[Y] = 0 $

The problem: we are asked to prove or give a counterexample to 1 leads to 2 and to 2 leads to 1

I have tried to prove these two directions but got nothing just by looking at the definitions and so I thought maybe they are false and we are to give counterexamples but I got nothing there either. Looking at the covariance formula I see Y is involved but I cannot really seem to incorporate it from conditional expectation given, so I am stuck and need help. Thanks to all.

- If $E(X\mid Y)=Z$, $E(Y\mid Z)=X$ and $E(Z\mid X)=Y$, then $X=Y=Z$ almost surely
- Conditional expectation to de maximum $E(X_1\mid X_{(n)})$
- 3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix
- Is it true that $\mathbb{E}+|\mathbb{E}\rvert\geq\mathbb{E}\rvert]+\mathbb{E}\rvert]$?
- Computing the expectation of conditional variance in 2 ways
- Calculation of covariances $cov(x_i^{2},x_j)$ and $cov(x_i^{2},x_j^{2})$ for multinomial distribution
- Fubini's theorem for conditional expectations
- Covariance between squared and exponential of Gaussian random variables
- Is a probability of 0 or 1 given information up to time t unchanged by information thereafter?
- Understanding the measurability of conditional expectations

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- How to Prove that if f(z) is entire, and f(z+i) = f(z), f(z+1) = f(z), then f(z) is constant?