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We have a matrix $$\left( \begin{array}{cc} X & X \\ -X & 1-X \\ \end{array} \right)$$ where $X$ is a random variable between $0$ and $1$. I heard about "random matrices". Is it an example of these matrices? Moreover, how we define a "mean" for these matrices? Since $X\in[0,1]$ I think that $$\left( \begin{array}{cc} 1/2 & 1/2 \\ -1/2 & 1/2 \\ \end{array} \right)$$ is the mean of $$\left( \begin{array}{cc} X & X \\ -X & 1-X \\ \end{array} \right)$$ What do you think?

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    $\begingroup$ It depends on the distribution of $X$, obviously. $X\in[0,1]$ does not imply $\mathbb{E}[X]=\frac{1}{2}$. $\endgroup$ – Jack D'Aurizio Jul 28 '15 at 14:48
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    $\begingroup$ If E(X)=.5 yes this the matrix of the means. Is this your question? $\endgroup$ – Did Jul 28 '15 at 14:49
  • $\begingroup$ @Did Yes, I'm getting closer now to this topic and I have many doubts. $\endgroup$ – Lely Jul 28 '15 at 15:00
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The mean is taken entrywise, i.e. $$ \mathbb E\pmatrix{X&X\\ -X&1-X}=\pmatrix{\mathbb EX&\mathbb EX\\ -\mathbb EX&1-\mathbb EX}. $$

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