연속확률변수의 분산

$g ( x ) = x f ( x )$라 하면 $g ( - x ) = - xf ( - x ) = - xf ( x ) = - g ( x )$ 이므로 $\displaystyle \int _{-1} ^{1} xf ( x ) dx = 0$ $\therefore$ $E ( X ) = 0$ $h ( x ) = x ^{2} f ( x )$라 하면 $h ( - x ) = ( - x ) ^{2} f ( - x ) = x ^{2} f ( x ) = h ( x )$ 이므로 $\displaystyle \int _{-1} ^{1} x ^{2} f ( x ) dx = 2 \int _{0} ^{1} x ^{2} f ( x ) dx = 2 \times \frac{1}{10} = \frac{1}{5}$ $\therefore$$\displaystyle V ( X ) = \int _{-1} ^{1} x ^{2} f ( x ) dx - \left\{ \int _{-1} ^{1} x f ( x ) dx \right\} ^{2}$ $\displaystyle = \frac{1}{5}$ $\therefore$ $\displaystyle V ( 10 X + 3 ) = 100 V ( X ) = 100 \times \frac{1}{5} = 20$