(0)

マルコフの不等式
\[ P\left(\left|X\right|\geq\epsilon\right)\leq\frac{E\left(\left|X\right|\right)}{\epsilon} \] で\(\epsilon\rightarrow\epsilon^{2}\quad,\quad X\rightarrow\left(X-\mu\right)^{2}\)とすると、
\begin{align*} P\left(\left(X-\mu\right)^{2}\geq\epsilon^{2}\right) & =P\left(\left|X-\mu\right|\geq\epsilon\right)\\ & \leq\frac{E\left(\left(X-\mu\right)^{2}\right)}{\epsilon^{2}}\\ & =\frac{V(X)}{\epsilon^{2}} \end{align*} これより、
\[ P(\left|X-\mu\right|\geq\epsilon)\leq\frac{V(X)}{\epsilon^{2}} \]

(0)-2

\begin{align*} V(X) & =\int_{-\infty}^{\infty}(x-\mu)^{2}P(x)dx\\ & =\int_{-\infty}^{\mu-\epsilon}(x-\mu)^{2}P(x)dx+\int_{\mu-\epsilon}^{\mu+\epsilon}(x-\mu)^{2}P(x)dx+\int_{\mu+\epsilon}^{\infty}(x-\mu)^{2}P(x)dx\qquad,\qquad\forall\epsilon>0\\ & \geq\int_{-\infty}^{\mu-\epsilon}(x-\mu)^{2}P(x)dx+\int_{\mu+\epsilon}^{\infty}(x-\mu)^{2}P(x)dx\\ & \geq\int_{u-\epsilon\geq x}\epsilon^{2}P(x)dx+\int_{\mu+\epsilon\leq x}\epsilon^{2}P(x)dx\\ & =\epsilon^{2}\int_{-x+\mu\geq\epsilon}P(x)dx+\epsilon^{2}\int_{x-\mu\geq\epsilon}P(x)dx\\ & =\epsilon^{2}\int_{\left|x-\mu\right|\geq\epsilon}P(x)dx\\ & =\epsilon^{2}P(\left|X-\mu\right|\geq\epsilon) \end{align*} これより、
\[ P(\left|X-\mu\right|\geq\epsilon)\leq\frac{V(X)}{\epsilon^{2}} \]