두 정규분포 비교 보기

$\displaystyle P ( | X | \leq a ) = P \left( \left| \frac{X - 0}{\sigma} \right| \leq \frac{a - 0}{\sigma} \right)$$\displaystyle = P \left( | Z | \leq \frac{a}{\sigma} \right)$ $\displaystyle P ( | Y | \leq B ) = P \left( \left| \frac{Y - 0}{\frac{\sigma}{2}} \right| \leq \frac{b - 0}{\frac{\sigma}{2}} \right)$$\displaystyle = P \left( | Z | \leq \frac{2 b}{\sigma} \right)$ 이므로 $P ( | X | \leq a ) = P ( | Y | \leq b )$이면 $a = 2 b$이다. ㄱ. $a = 2 b$이고 $a > 0 , b > 0$이므로 $a > b$ (참) ㄴ. $\displaystyle P \left( Y > \frac{a}{2} \right) = P \left( \frac{Y - 0}{\frac{\sigma}{2}} > \frac{b - 0}{\frac{\sigma}{2}} \right)$$\displaystyle = P \left( Z > \frac{2 b}{\sigma} \right)$ (참) ㄷ. $P ( Y > b ) = 1 - P ( Y \leq b ) = 0.3$이므로 $P ( | Y | \leq b ) = 1 - P ( | Y | > b )$$= 1 - 2 P ( Y > b )$$= 1 - 2 \times 0.3 = 0.4$ ∴ $\left. P ( | X | \leq a \right) = P ( | Y | \leq b ) = 0.4$ (거짓)