확률밀도함수 확률

확률밀도함수의 정의와 삼각형의 닮음으로 부터

$\displaystyle \mathrm{P} \mathit{\left( 0 \leq X \leq \frac{1}{2} \right)} = \left( \frac{1}{4} \right) ^{2} \mathrm{P} \mathit{(} 0 \leq X \leq 2 )$$\displaystyle = \frac{1}{16} \times \frac{1}{2} = \frac{1}{32}$ $\displaystyle {\mathrm{P}} ( 3 \leq X \leq 4 ) = \left( \frac{1}{2} \right) ^{2} {\mathrm{P}} ( 2 \leq X \leq 4 )$$\displaystyle = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$ $\therefore$ $\displaystyle {\mathrm{P}} \left( \frac{1}{2} \leq X \leq 3 \right) = 1 - \frac{1}{32} - \frac{1}{8} = \frac{27}{32}$