삼각형 확률밀도함수

$\displaystyle \mathrm{P} ( \mathit{0} \leq X \leq a ) = \frac{1}{2} ab = 1$ $\therefore ab = 2 \cdots \cdots \text{㉠}$ (i) $0 < a \leq 4$일 때 $\displaystyle \mathrm{P} \left( \mathit{0} \leq X \leq \frac{a}{2} \right) = \frac{1}{2} \cdot \frac{a}{2} \cdot \left( \frac{b}{2} \cdot \frac{a}{2} \right) = \frac{b}{2}$ ㉠에 의해 $\displaystyle \frac{a}{8} = \frac{b}{2}$ $\therefore a = 4 b$ ㉠에서 $4 b ^{2} = 2$ $\displaystyle \therefore b = \frac{1}{\sqrt{2}}$ $\displaystyle \therefore a = 2 \sqrt{2}$ $\therefore a ^{2} + 4 b ^{2} = 8 + 2 = 10$ (ii) $a > 4$일 때는 부적합하다.