# Mean, Median and Standard Deviation

Discrete Mathematics Level pending

Consider the set of all probability distributions $$p$$ (which may be either discrete or continuous) over the real line $$\mathbb{R}$$, with mean $$\mu= 10$$ and standard deviation $$\sigma = 5$$. As usual, define a median of the distribution to be any real number $$m$$ such that
$\mathbb{P}_{X\sim p} (X\leq m) \geq \frac{1}{2}, \hspace{5pt} \mathbb{P}_{X\sim p} (X\geq m) \geq \frac{1}{2}.$ Find the maximum possible value of $$m$$.

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