Given three random variables $X, Y$ and $Z$ which are $N(m,\sigma^2), N(\mu,\sigma^2)$ and $N(m+\mu,\sigma^2)$ respectively for some $m, \mu, \sigma$, does it follow that $X, Y, Z$ are mutually independent?
2026-03-25 16:05:09.1774454709
Mutual independence, normal distribution
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No. Take for example $X\sim \mathcal N(0,1)$ and $Y=\mu+X$. Then $X$ and $Y$ has the whiched property but they are not independent.