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  1. Let $X$ be a uniform random vector. Is any linear transformation $AX$ of $X$ still uniformly distributed? I know it is yes when $A$ is square and invertible, by using the change of variable formula. But not sure when $A$ is square and not invertible, or $A$ is not square.
  2. If $X$ and $Y$ are both uniform random variables (or vectors), will $(X^T,Y^T)^T$ be a random vector?

Thanks!

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    Yes, I am asking what your definition of uniform random vector is. You gave one definition and then a different one. Which one is it?2012-10-26

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Try the case $A = \pmatrix{1 & 1\cr 0 & 0\cr}$ or the $1 \times 2$ matrix $A = (1,1)$.

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    Thanks! (1) The examples explain the answer to the first part is no. So I guess the correct statement should be any invertible linear transformation of a uniform random vector is still uniformly distributed? (2) How about the second part? I guess it is no too, because it seems possible to construct a distribution in a 2D shape, such that the projections to both axes will be uniform distributions.2012-10-25