확률밀도함수 확률

$0 \leq x \leq 1$에서 정의된 확률밀도함수 $f \left( x \right) = k x \left( 1 - x \right)$에 대하여 $\displaystyle \int _{0} ^{1} {} f \left( x \right) dx = 1$ $\displaystyle \int _{0} ^{1} {} f \left( x \right) dx$$\displaystyle = \int _{0} ^{1} {} \left( kx - kx ^{2} \right) dx$$\displaystyle = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} \frac{k}{2} x ^{2} - \frac{k}{3} x ^{3} \end{array} \right] _{0} ^{1}$ $\displaystyle = \frac{k}{2} - \frac{k}{3}$$\displaystyle = \frac{k}{6} = 1$ $\therefore$$k = 6$ $\therefore$$f \left( x \right) = 6 x \left( 1 - x \right) = 6 x - 6 x ^{2}$ $\displaystyle {\mathrm{P}} \left( \left| X - \frac{1}{2} \right| \leq \frac{1}{4} \right)$$\displaystyle = \mathrm{P} \mathit{\left( - \frac{1}{4} \leq X - \frac{1}{2} \leq \frac{1}{4} \right)}$$\displaystyle = \mathrm{P} \mathit{\left( \frac{1}{4} \leq X \leq \frac{3}{4} \right)}$ $\therefore$$\displaystyle \mathrm{P} \mathit{\left( \frac{1}{4} \leq X \leq \frac{3}{4} \right)} = \int _{\frac{1}{4}} ^{\frac{3}{4}} {} \left( 6 x - 6 x ^{2} \right) dx$$\displaystyle = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} 3 x ^{2} - 2 x ^{3} \end{array} \right] _{\frac{1}{4}} ^{\frac{3}{4}}$$\displaystyle = \frac{11}{16}$