Existence of expectation

• Definition: A random variable \(X\) has an expectation if is absolutely integrable:

\[\mathbb{E}[ |X| ] = \int\limits_{\Omega} |x| f_X(x)\; dx < \infty\]

• Theorem: Any random variable \(X\) satisfies \(| \mathbb{E}[X]| \leq \mathbb{E}[|X|]\) .

• Example: A random variable whose expectation is undefined is the Cauchy random variable: \(f_X(x) = \dfrac{1}{\pi (1+x^2)}\), for \(x\in\mathbb{R}\).

\[\begin{align} \mathbb{E}[|X|] &= \int\limits_{-\infty}^{\infty} |x| \dfrac{1}{\pi (1+x^2)}\; dx \\ &= 2 \cdot \int\limits_{0}^{\infty} \dfrac{x}{\pi (1+x^2)}\; dx \\ &\geq 2 \cdot \int\limits_{1}^{\infty} \dfrac{x}{\pi (1+x^2)}\; dx \\ &\geq 2 \cdot \int\limits_{1}^{\infty} \dfrac{x}{\pi (x^2+x^2)}\; dx \\ &= \dfrac{1}{\pi}(\log(x))\Bigg{|}_{1}^{\infty}= \infty \;. \end{align}\]