Let \(X\) be any absolutely continuous random variable with pdf \(f\) and cdf \(F\), and assume that \(E[(X-a)^2]\) and \(E\left[|X-a|\right]\) are finite for all \(a\in \mathbb{R}\).
- Compute and interpret
\[ a_0=\underset{a\in\mathbb{R}}{\arg\min}\,E[(X-a)^2]. \]
- Compute and interpret
\[ b_0=\underset{b\in\mathbb{R}}{\arg\min}\,E\left[|X-b|\right]. \]