정규분포 대칭 조건

확률변수 $X$의 확률밀도함수를 $f \left( x \right)$라 하면 곡선 $y = f \left( x \right)$는 직선 $x = m$에 대하여 대칭이다. ${\mathrm{P}} \left( X \leq a \right) + {\mathrm{P}} \left( X \leq a ^{2} \right) = 1$에서 $a ^{2} + a = 2 m$ ${\mathrm{P}} \left( X \leq a ^{2} + a \right)$$\displaystyle = {\mathrm{P}} \left( X \leq 2 m \right) = {\mathrm{P}} \left( Z \leq \frac{2 m - m}{\frac{1}{2 m}} \right)$ $= {\mathrm{P}} \left( Z \leq 2 m ^{2} \right) = 0.9772$ ${\mathrm{P}} \left( Z \leq 2 \right) = 0.9772$이므로 $2 m ^{2} = 2$ 표준편차 $\displaystyle \frac{1}{2 m}$은 양수이므로 $m = 1$ $a ^{2} + a = 2$, $\left( a + 2 \right) \left( a - 1 \right) = 0$에서 $a < 0$이므로 $a = - 2$ $\displaystyle {\mathrm{P}} \left( X \leq - \frac{a}{8} \right)$$\displaystyle = {\mathrm{P}} \left( X \leq \frac{1}{4} \right) = {\mathrm{P}} \left( Z \leq \frac{\frac{1}{4} - 1}{\frac{1}{2}} \right)$ $= {\mathrm{P}} \left( Z \leq - 1.5 \right)$ $= 0.5 - {\mathrm{P}} \left( - 1.5 \leq Z \leq 0 \right)$ $= 0.5 - {\mathrm{P}} \left( 0 \leq Z \leq 1.5 \right)$ $= 0.5 - 0.4332 = 0.0668$