정규분포 대칭 평균 분산

확률변수 $X$는 정규분포 $\mathrm{N} \mathit{(} m , \sigma ^{2} )$을 따르고 $\mathrm{P} \mathit{(} X < a - 3 ) = \mathrm{P} \mathit{(} X > b + 2 )$이므로 $\displaystyle \frac{( a - 3 ) + ( b + 2 )}{2} = m$ $\displaystyle m = \frac{a + b - 1}{2}$ …① $\displaystyle Y = \frac{1}{3} X + 1$, ${\mathrm{E}} ( Y ) = 51$이므로 $\displaystyle {\mathrm{E}} ( Y ) = {\mathrm{E}} ( \frac{1}{3} X + 1 ) = \frac{1}{3} m + 1$ $m = 150$ … ② ①, ② 에 의하여 $a + b = 301$ $\displaystyle {\mathrm{V}} ( Y ) = {\mathrm{V}} ( \frac{1}{3} X + 1 ) = \frac{1}{9} {\mathrm{V}} ( X )$ $\displaystyle \frac{1}{9} {\mathrm{V}} ( X ) = \frac{4}{9}$이므로 ${\mathrm{V}} ( X ) = 4$, $\sigma = 2$ $\therefore$ $a + b + \sigma = 303$