Minimizer of f(z) = E(|X-z|) is median

Dependencies:

1. Median of a random variable
2. Random variables: multiple medians
3. Expected value of a random variable
4. Linearity of expectation
5. /measure-theory/lebesgue-integral

$\newcommand{\E}{\operatorname{E}}$ Let $X$ be a real-valued random variable (over probability space $(\Omega, \mathcal{F}, \Pr)$). Let $f(z) = \E(|X - z|)$. Then

1. If $m$ is a median of $X$ and $c$ is not a median of $X$, then $f(c) > f(m)$.
2. If $m_1$ and $m_2$ are medians of $X$, then $f(m_1) = f(m_2)$.

Proof

Let $a < b$. Then \begin{align} & f(b) - f(a) = \E(|X-b|) - \E(|X-a|) \\ &= \E(|X-b|-|X-a|) \tag{linearity of expectation} \\ &= \int_{\omega \subseteq \Omega} (|X(\omega)-b|-|X(\omega)-a|)\Pr(\omega) \\ &= \int_{\omega \subseteq \Omega \wedge X(\omega) \le a} (|X(\omega)-b|-|X(\omega)-a|)\Pr(\omega) \\ &\qquad+ \int_{\omega \subseteq \Omega \wedge a < X(\omega) < b} (|X(\omega)-b|-|X(\omega)-a|)\Pr(\omega) \\ &\qquad+ \int_{\omega \subseteq \Omega \wedge X(\omega) \ge b} (|X(\omega)-b|-|X(\omega)-a|)\Pr(\omega) \\ &= \int_{\omega \subseteq \Omega \wedge X(\omega) \le a} (b-a)\Pr(\omega) \\ &\qquad+ \int_{\omega \subseteq \Omega \wedge a < X(\omega) < b} (a+b-2X(\omega))\Pr(\omega) \\ &\qquad+ \int_{\omega \subseteq \Omega \wedge X(\omega) \ge b} (a-b)\Pr(\omega) \\ &= (b-a)(\Pr(X \le a) - \Pr(X \ge b)) + \int_{\omega \subseteq \Omega \wedge a < X(\omega) < b} (a+b-2X(\omega))\Pr(\omega). \end{align}

Proof of part 2

Let $m_1$ and $m_2$ be medians of $X$. Without loss of generality, assume $m_1 < m_2$. Then $\Pr(m_1 < X < m_2) = 0$, $\Pr(X \le m_1) = 1/2$ and $\Pr(X \ge m_2)$. Hence, \[ f(m_2) - f(m_1) = (m_2 - m_1)(\Pr(X \le m_1) - \Pr(X \ge m_2)) = 0. \]

Proof of part 1 when $c > m$

Since $m$ is a median, we get $\Pr(X \le c) \ge \Pr(X \le m) \ge 1/2$. Since $c$ is not a median, we get $\Pr(X \ge c) < 1/2$.

Case 1: $\Pr(m < X < c) = 0$.

\[ f(c) - f(m) = (c-m)(\Pr(X \le m) - \Pr(X \ge c)) > (c-m)(1/2 - 1/2) = 0 \]

Case 2: $\Pr(m < X < c) > 0$.

\[ m < X < c \implies -(c-m) < m+c-2X < c-m \] Hence, \begin{align} & f(c) - f(m) \\ &= (c-m)(\Pr(X \le m) - \Pr(X \ge c)) + \int_{\omega \subseteq \Omega \wedge m < X(\omega) < c} (m+c-2X(\omega))\Pr(\omega). \\ &> (c-m)(\Pr(X \le m) - \Pr(X \ge c)) - (c-m)\Pr(m < X < c) \\ &= (c-m)(\Pr(X \le m) - \Pr(X > m)) \\ &= (c-m)(2\Pr(X \le m) - 1) \ge 0 \tag{$m$ is a median} \end{align}

Proof of part 1 when $c < m$

(Analogous to the case where $c > m$.)

Dependency for:

1. |mean - median| ≤ stddev

Info:

• Depth: 8
• Number of transitive dependencies: 22

Transitive dependencies:

1. /analysis/topological-space
2. /sets-and-relations/countable-set
3. /sets-and-relations/de-morgan-laws
4. /measure-theory/linearity-of-lebesgue-integral
5. /measure-theory/lebesgue-integral
6. σ-algebra
7. Generated σ-algebra
8. Borel algebra
9. Measurable function
10. Generators of the real Borel algebra (incomplete)
11. Measure
12. σ-algebra is closed under countable intersections
13. Group
14. Ring
15. Field
16. Vector Space
17. Probability
18. Random variable
19. Median of a random variable
20. Random variables: multiple medians
21. Expected value of a random variable
22. Linearity of expectation