확률밀도함수

확률밀도함수의 정의된 구간 내에서의 $x$축으로 둘러싸인 넓이가 $1$이므로 $\displaystyle \int _{-1} ^{0} a \left( 1 - x ^{2} \right) dx + \int _{0} ^{3} a \left( 1 - \frac{x}{3} \right) dx$ $\displaystyle = a \left[ x - \frac{1}{3} x ^{3} \right] _{-1} ^{0} + a \left[ x - \frac{1}{6} x ^{2} \right] _{0} ^{3}$ $\displaystyle = \frac{2}{3} a + \frac{3}{2} a = \frac{13}{6} a = 1$ $\displaystyle \therefore a = \frac{6}{13}$ $\displaystyle \therefore P ( - 1 \leq X \leq 0 ) = \frac{2}{3} a = \frac{2}{3} \times \frac{6}{13} = \frac{4}{13}$ $\therefore p + q = 17$