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Consider a subgaussian random variable $X$,

$$P[ |X - \mathbb E X | > t ] < C_1 \exp (-\frac{t^2}{K}) $$

for $t>0$ and constants $K, C_1$. Define a median $m$ as a value such that

$$P[ X \geq m ] = P[X \leq m] = \frac{1}{2}.$$

Show that

$$ |\mathbb E X - m | <C_2 K $$ for constant $C_2$.

I thought the solution would involve rewriting thusly,

\begin{align*} |\mathbb E X - m | \leq \mathbb E | X - m| = \int_{0}^{\infty} P[ |X-m| > t]dt, \end{align*}

and somehow using the definition of the median. But I can't seem to rearrange it in a way that is amenable. Any ideas would be appreciated.

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1 Answer 1

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You could use $\sqrt{\textrm{Var}(X)} \geq |\mathbb{E}X-\mathbb{M}X|$ and bound the variance of $X$ by the proxy variance.

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