두 정규분포 비교

두 확률변수 $X$, $Y$가 정규분포를 따르므로 $Z$가 표준정규분포를 따르는 확률변수일 때, $\mathrm{P} \mathit{\left( X \leq 0 \right)} = \mathrm{P} \mathit{\left( Z \leq - m \right)}$ $\displaystyle \mathrm{P} \mathit{\left( Y \leq 0 \right)} = \mathrm{P} \mathit{\left( Z \leq \frac{- \left( m ^{2} + 2 m + 16 \right)}{\sigma} \right)}$ $\mathrm{P} \mathit{\left( X \leq 0 \right)} = \mathrm{P} \mathit{\left( Y \leq 0 \right)}$에서 $\displaystyle - m = \frac{- \left( m ^{2} + 2 m + 16 \right)}{\sigma}$ $\displaystyle \sigma = \frac{m ^{2} + 2 m + 16}{m} = m + 2 + \frac{16}{m}$ $m > 0$, $\displaystyle \frac{16}{m} > 0$이므로 절대부등식의 성질에 의하여 $\displaystyle \sigma \geq 2 + 2 \sqrt{m \times \frac{16}{m}} = 10$ $\displaystyle m = \frac{16}{m}$일 때, $\sigma$의 값이 최소이므로 $m _{1} = 4$ $m = 4$일 때, 두 확률변수 $X$, $Y$는 각각 정규분포 $\mathrm{N} \mathit{\left( 4 , 1 ^{2} \right)}$, $\mathrm{N} \mathit{\left( 40 , 10 ^{2} \right)}$을 따르므로 $\displaystyle \mathrm{P} \mathit{\left( X \geq 1 \right)} = \mathrm{P} \mathit{\left( Z \geq \frac{1 - 4}{1} \right)}$$= \mathrm{P} \mathit{\left( Z \geq - 3 \right)}$ $\displaystyle \mathrm{P} \mathit{\left( Y \leq k \right)} = \mathrm{P} \mathit{\left( Z \leq \frac{k - 40}{10} \right)}$ $\mathrm{P} \mathit{\left( X \geq 1 \right)} = \mathrm{P} \mathit{\left( Y \leq k \right)}$에서 $\mathrm{P} \mathit{\left( Z \geq - 3 \right)}$$\displaystyle = \mathrm{P} \mathit{\left( Z \leq \frac{k - 40}{10} \right)}$$\displaystyle = \mathrm{P} \mathit{\left( Z \geq - \frac{k - 40}{10} \right)}$ $- 3$$\displaystyle = - \frac{k - 40}{10}$ 따라서 $k = 70$