If $x, y, z $ are independent random variables, such as all following a uniform distribution between $0$ and $1$, are $x-y, y-z, ,x-z$ also independent of each other?
2026-03-29 20:06:41.1774814801
If $x, y, z$ are independent random variables, are $ x-y, y-z, x-z$ also independent of each other?
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We have $$ \mathbb{E}[(x-y)(y-z)]=-\mathbb{E}[y^2]-\mathbb{E}[x]\mathbb{E}[y]-\mathbb{E}[x]\mathbb{E}[z]+\mathbb{E}[y]\mathbb{E}[z] $$ and similarly for the other products. So the r.v. are independent iif $$ \mathbb{E}[y^2]=\left(\mathbb{E}[y]\right)^2 $$ i.e. iif the r.v. have zero variance. This is not the case for the uniform distribution