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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