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Let $a \in (0,1)$ and $X \sim \text{Unif}[0,a]$ we could generate a random variable $Y$ with conditional distribution $$(1-a)\delta_0 +a \delta_{X/a}, \ \ \ \ \text{given} \ \ X.$$ Then, $$\mathbb{E}(Y|X) \equiv X$$ and $$Y \sim (1-a)\delta_0 + a \text{Unif}[0,1]$$

My question is: can we generate a random variable $\tilde{Y}$ instead of Dirac measure 0 to have 1 and which would satisfy $$\mathbb{E}(Y|X) \equiv X$$ because I don't know for what kind of $X \sim Unif[?,?]$ this would be satisfied. It doesn't mean to have also same form, but it should have a point mass (Dirac measure) and the rest some uniform distribution on [0,1]$.

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