For random variable $X$: $\mathbb{E}|X| < \infty$ prove that $\mathbb{E}|X − Med X| = \inf{ \{ \mathbb{E}|X − a| : a \in \mathbb{R} \} }$

Question

For random variable $X$ we know that: $\mathbb{E}|X| < \infty$. Prove that for every $a \in \mathbb{R}$ it is true that:

1. $\mathbb{E}|X − a| < \infty$
2. $\mathbb{E}|X − Med X| = \inf{ \{ \mathbb{E}|X − a| : a \in \mathbb{R} \} } \text{ where Med is a median }$

For the first part:

In general we know that: $ \mathbb{E}|X−a| \leq \mathbb{E}(|X|+|a|)$

From linearity of expectation we know that: $\ \mathbb{E}|X|+\mathbb{E}|a|$

Therefore: $\mathbb{E}|X−a| \leq \mathbb{E}|X| + \mathbb{E}|a|$ and from here:

• form assumption we know that: $\mathbb{E}|X| < \infty$
• in general we know that: $\mathbb{E}|a|=|a|$ and $ |a| < \infty$ for $a \in \mathbb{R}$

However, I don't know how to prove the second part. Any help would be much appreciated.