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Let $\mathbf{x}=[x_i]_{i=1}^d, \mathbf{y}=[y_i]_{i=1}^d$ be two vectors in $R^d$.

Is it possible to find a lower bound $l\leq \|x-y\|$ and an upper bound $u\geq\|x-y\|$ as a function of $\bar{\mathbf{x}}, \bar{\mathbf{y}}, \sigma({\mathbf x)}$ and $\sigma({\mathbf y})$ ,

where ${\bar v} = \frac{1}{d}\sum_{i=1}^d{v_i}$ and $\sigma(v)=\sqrt{\frac{1}{d}\sum_{i=1}^d{(v_i-\bar v)^2}}$ (the mean and the standard the deviation of $\mathbf{v}$)?

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    What do you mean by mean and standard deviation of two vectors? Don't you mean random variables (vector-valued)? In any case, I don't think so. There is no way to tell if they're equal even if you know the means and standard deviations are the same.2012-11-14

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