정규분포 기댓값

${\mathrm{E}} ( X ) = m$, $\sigma ( X ) = \sigma$라 하면 조건 (가)에 의하여 ${\mathrm{E}} ( Y ) = a m$, $\sigma ( Y ) = a \sigma$ (∵ $a > 0$) 조건 (나)에서 ${\mathrm{P}} ( X \leq 18 ) + {\mathrm{P}} ( Y \geq 36 ) = 1$ $\leftrightarrow$ $\displaystyle {\mathrm{P}} \left( Z \leq \frac{18 - m}{\sigma} \right) + {\mathrm{P}} \left( Z \geq \frac{36 - am}{a \sigma} \right) = 1$ $\leftrightarrow$ $\displaystyle \frac{18 - m}{\sigma} = \frac{36 - am}{a \sigma}$ ∴ $a = 2$ (∵ $\sigma > 0$) 조건 (다)에서 ${\mathrm{P}} ( X \leq 28 ) = {\mathrm{P}} ( Y \geq 28 )$ $\leftrightarrow$ $\displaystyle {\mathrm{P}} \left( Z \leq \frac{28 - m}{\sigma} \right) = {\mathrm{P}} \left( Z \geq \frac{28 - 2 m}{2 \sigma} \right)$ (∵ $a = 2$) $\leftrightarrow$ $\displaystyle \frac{28 - m}{\sigma} = - \frac{28 - 2 m}{2 \sigma}$ ∴ $m = 21$ (∵ $\sigma > 0$) ∴ ${\mathrm{E}} ( Y ) = am = 42$