題組內容

3. Let \( X \) and \( Y \) be random variables and \( \{X_n, n \ge 1\} \) be a sequence of random variables. Define \[ \rho(X) = E\left( \frac{|X|}{1+|X|} \right). \] Show that

(b) $\rho(\sigma X) \le \max\{|\sigma|, 1\}\rho(X)$, where $\sigma$ is a real number, (5%)