확률밀도함수

$\left\{ g \left( x \right) - f \left( x \right) \right\} \left\{ g \left( x \right) - a \right\} = 0$이므로 $g \left( x \right) = f \left( x \right)$ 또는 $g ( x ) = a$ 조건 (가)와 (나)에 의하여 확률밀도함수 $y = g ( x )$의 그래프는 그림과 같다.

$\mathrm{P} \mathit{\left( 0 \leq Y \leq 4 \right)} = 1$이므로 $\displaystyle \frac{1}{2} \times 2 a \times a + \left( 4 - 4 a \right) \times a + \frac{1}{2} \times 2 a \times a = 1$ $2 a ^{2} - 4 a + 1 = 0$, $\displaystyle a = \frac{2 \pm \sqrt{2}}{2}$ $\displaystyle 0 < a < \frac{1}{2}$이므로 $\displaystyle a = \frac{2 - \sqrt{2}}{2}$, $1 < 5 a < 2$ $\mathrm{P} \mathit{\left( 0 \leq Y \leq 5 a \right)}$ $= \mathrm{P} \mathit{\left( 0 \leq Y \leq 2 a \right)} + \mathrm{P} \mathit{\left( 2 a \leq Y \leq 5 a \right)}$ $\displaystyle = \frac{1}{2} \times 2 a \times a + 3 a \times a = 4 a ^{2}$ $\displaystyle = 4 \times \left( \frac{2 - \sqrt{2}}{2} \right) ^{2} = 6 - 4 \sqrt{2}$ 따라서 $p = 6$, $q = 4$이므로 $p \times q = 24$