확률밀도함수 확률

함수 $g ( x )$가 구간 $[ 0 , 2 ]$에서 정의된 연속확률변수 $X$의 확률밀도함수이므로 ${\mathrm{P}} ( 0 \leq X \leq 2 )$$=$$\displaystyle \int _{0} ^{2} g ( x ) dx$$= 1$이다. $\displaystyle \int _{0} ^{2} g ( x ) dx$$\displaystyle = \int _{0} ^{1} g ( x ) dx$$+$$\displaystyle \int _{1} ^{2} g ( x ) dx$ $\displaystyle = \int _{0} ^{1} f ( x ) dx + \int _{1} ^{2} \left\{ f ( x - 1 ) + f ( 1 ) \right\} dx$ $\displaystyle = \int _{0} ^{1} f ( x ) dx + \int _{0} ^{1} \left\{ f ( x ) + f ( 1 ) \right\} dx$ $\displaystyle = 2 \int _{0} ^{1} f ( x ) dx + f ( 1 )$$\displaystyle = 2 \int _{0} ^{1} ax ^{2} dx + a$ $\displaystyle = 2 \left[ \frac{ax ^{3}}{3} \right] _{0} ^{1} + a$ $\displaystyle = \frac{5}{3} a$ 따라서 $\displaystyle \frac{5}{3} a = 1$에서 $\displaystyle a = \frac{3}{5}$이다. ∴ $\displaystyle f ( x ) = \frac{3}{5} x ^{2}$ ∴ $\displaystyle {\mathrm{P}} \left( \frac{3}{5} \leq X \leq \frac{8}{5} \right)$$\displaystyle = \int _{\frac{3}{5}} ^{\frac{8}{5}} g ( x ) dx$ $\displaystyle = \int _{\frac{3}{5}} ^{1} g ( x ) dx + \int _{1} ^{\frac{8}{5}} g ( x ) dx$ $\displaystyle = \int _{\frac{3}{5}} ^{1} f ( x ) dx + \int _{1} ^{\frac{8}{5}} \left\{ f ( x - 1 ) + f ( 1 ) \right\} dx$ $\displaystyle = \int _{\frac{3}{5}} ^{1} f ( x ) dx + \int _{0} ^{\frac{3}{5}} \left\{ f ( x ) + f ( 1 ) \right\} dx$ $\displaystyle = \int _{0} ^{1} f ( x ) dx + \frac{3}{5} f ( 1 )$ $\displaystyle = \int _{0} ^{1} \frac{3}{5} x ^{2} dx + \left( \frac{3}{5} \right) ^{2}$ $\displaystyle = \left[ \frac{x ^{3}}{5} \right] _{0} ^{1} + \frac{9}{25}$$\displaystyle = \frac{14}{25}$ ∴ $p = 25$, $q = 14$ ∴ $p + q = 39$