이항분포 누적확률 함수

확률변수 $X$는 이항분포 $\displaystyle {\mathrm{B}} \left( 100 , \frac{1}{2} \right)$을 따르고 $f ( x ) = \mathrm{P} \mathit{(} X \leq 5 x + 50 ) = \mathrm{P} \mathit{(} X \leq [ 5 x + 50 ] )$ $\displaystyle = \sum\limits _{n = 0} ^{[ 5 x + 50 ]} {}_{100} {\mathrm{C}} _{n} \left( \frac{1}{2} \right) ^{100}$ (단, $[ 5 x + 50 ]$은 $5 x + 50$을 넘지 않은 최대정수) ㄱ. $\displaystyle V \left( X \right) = 100 \frac{1}{2} \left( 1 - \frac{1}{2} \right) = 25$ (참) ㄴ. $x _{1} \leq x _{2}$이면 $[ 5 x _{1} + 50 ] \leq [ 5 x _{2} + 50 ]$이므로 $\displaystyle f ( x _{1} ) = \sum\limits _{n = 0} ^{[ 5 x _{1} + 50 ]} {}_{100} {\mathrm{C}} _{n} \left( \frac{1}{2} \right) ^{100}$$\displaystyle \leq \sum\limits _{n = 0} ^{[ 5 x _{2} + 50 ]} {}_{100} {\mathrm{C}} _{n} \left( \frac{1}{2} \right) ^{100}$$= f ( x _{2} )$ (참) ㄷ. $- 10 \leq x \leq 10$인 임의의 $x$에 대하여, $f ( - x ) = {\mathrm{P}} \mathit{(} X \leq - 5 x + 50 )$ $\displaystyle = \sum\limits _{n = 0} ^{[ - 5 x + 50 ]} {}_{100} {\mathrm{C}} _{n} \left( \frac{1}{2} \right) ^{100}$ $\displaystyle \geq \sum\limits _{n = [ 5 x + 50 ] + 1} ^{100} {}_{100} {\mathrm{C}} _{n} \left( \frac{1}{2} \right) ^{100}$ $f ( x ) + f ( - x ) \geq$ $\displaystyle \sum\limits _{n = 0} ^{[ 5 x + 50 ]} {}_{100} {\mathrm{C}} _{n} \left( \frac{1}{2} \right) ^{100}$$+$$\displaystyle \sum\limits _{n = [ 5 x + 50 ] + 1} ^{100} {}_{100} {\mathrm{C}} _{n} \left( \frac{1}{2} \right) ^{100}$$= 1$ (거짓) 예를 들어 (ⅰ) $x = 1$인 경우 $f ( - 1 ) = \mathrm{P} \mathit{(} X \leq 45 )$$\displaystyle = \sum\limits _{n = 0} ^{45} {}_{100} {\mathrm{C}} _{n} \left( \frac{1}{2} \right) ^{100}$$\displaystyle = \sum\limits _{n = 55} ^{100} {}_{100} {\mathrm{C}} \mathit{_{n}} \left( \frac{1}{2} \right) ^{100}$ $f ( 1 ) + f ( - 1 )$$\displaystyle = \sum\limits _{n = 0} ^{55} {}_{100} {\mathrm{C}} \mathit{\mathrm{_{n}}} \left( \frac{1}{2} \right) ^{100}$$\displaystyle + \sum\limits _{n = 55} ^{100} {}_{100} \mathrm{C} \mathit{\mathrm{_{n}}} \left( \frac{1}{2} \right) ^{100} > 1$ (ⅱ) $x = 0.9$인 경우 $f ( - 0.9 ) = \mathrm{P} \mathit{(} X \leq 45 )$$\displaystyle = \sum\limits _{n = 0} ^{45} {}_{100} {\mathrm{C}} _{n} \left( \frac{1}{2} \right) ^{100}$ $\displaystyle = \sum\limits _{n = 55} ^{100} {}_{100} {\mathrm{C}} \mathit{\mathrm{_{n}}} \left( \frac{1}{2} \right) ^{100}$ $f ( 0.9 ) + f ( - 0.9 )$$\displaystyle = \sum\limits _{n = 0} ^{54} {}_{100} \mathrm{C} \mathit{\mathrm{_{n}}} \left( \frac{1}{2} \right) ^{100}$$\displaystyle + \sum\limits _{n = 55} ^{100} {}_{100} \mathrm{C} \mathit{\mathrm{_{n}}} \left( \frac{1}{2} \right) ^{100} = 1$