Solved – Does uniform conditional distribution imply independence

Take two random variable $X,Y$ and suppose $X$ is distributed uniformly on $[0,1]$ conditional on $Y$. Does this imply that $X$ is independent of $Y$? Could you make an example?

Independence would mean that knowing the value of $Y$ gives no information on the value of $X$.

So here $X$ will be independent of $Y$ only if $X$ has a uniform marginal distribution on $[0,1]$, and the conditional distribution $X|Y$ is uniform on $[0,1]$ independent of the value of $Y$.

An example be a uniform (joint) distribution over the unit square.

Here are some examples using Tetris blocks:

For the "S" block

Tetris "S" block (rotated)

we have $p[X|Y=text{middle}]=p[X]=text{uniform}$, but $X$ is certainly not independent of $Y$.

While for the "O" block

Tetris "O" block

we have $p[X|Y]=p[X]=text{uniform}$, so $X$ is independent of $Y$.

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