(a)離散型確率変数

\begin{align*} E\left(\left|X\right|\right) & =\sum_{i}\left|x_{i}\right|P\left(X=x_{i}\right)\\ & \geq\sum_{\left|x_{1}\right|\geq\epsilon}\left|x_{i}\right|P\left(X=x_{i}\right)\qquad,\qquad\forall\epsilon>0\\ & \geq\sum_{\left|x_{1}\right|\geq\epsilon}\epsilon P\left(X=x_{i}\right)\\ & =\epsilon P\left(\left|X\right|\geq\epsilon\right) \end{align*}

(b)連続型確率変数

\begin{align*} E\left(\left|X\right|\right) & =\int_{-\infty}^{\infty}\left|x\right|P(x)dx\\ & =\int_{-\infty}^{-\epsilon}\left|x\right|P(x)dx+\int_{-\epsilon}^{\epsilon}\left|x\right|P(x)dx+\int_{\epsilon}^{\infty}\left|x\right|P(x)dx\qquad,\qquad\forall\epsilon>0\\ & \geq\int_{-\infty}^{-\epsilon}\epsilon P(x)dx+\int_{\epsilon}^{\infty}\epsilon P(x)dx\\ & =\epsilon\int_{-\epsilon\geq x}P(x)dx+\epsilon\int_{\epsilon\leq x}P(x)dx\\ & =\epsilon\int_{-x\geq\epsilon}P(x)dx+\epsilon\int_{x\geq\epsilon}P(x)dx\\ & =\epsilon\int_{\left|x\right|\geq\epsilon}P(x)dx\\ & =\epsilon P\left(\left|X\right|\geq\epsilon\right) \end{align*}

(*)

これより、
\[ P\left(\left|X\right|\geq\epsilon\right)\leq\frac{E\left(\left|X\right|\right)}{\epsilon} \]