확률밀도함수 판정

$[ 0 , 1 ]$에서 $f ( x )$와 $g ( x )$가 확률밀도함수이므로 $f ( x ) \geq 0 , g ( x ) \geq 0$이고$\displaystyle \int _{0} ^{1} f ( x ) dx = \int _{0} ^{1} g ( x ) dx = 1$ ① $\displaystyle \int _{0} ^{1} \left\{ f ( x ) - g ( x ) \right\} dx = 0$ ② $\displaystyle \int _{0} ^{1} \left\{ f ( x ) + g ( x ) \right\} dx = 2$ ③ $\displaystyle \int _{0} ^{1} \frac{1}{2} \left\{ f ( x ) - g ( x ) \right\} dx = 0$ ④ $\displaystyle \int _{0} ^{1} \frac{1}{3} \left\{ 2 f ( x ) + g ( x ) \right\} dx$ $\displaystyle = \frac{2}{3} \int _{0} ^{1} f ( x ) dx + \frac{1}{3} \int _{0} ^{1} g ( x ) dx$$\displaystyle = \frac{2}{3} + \frac{1}{3} = 1$<참> ⑤ $\displaystyle \int _{0} ^{1} \left\{ 2 f ( x ) - g ( x ) \right\} dx = 1$이지만 $2 f ( x ) - g ( x ) < 0$일 수도 있다.