이산확률변수의 분산

이산확률변수 $X$가 가지는 값은 $0$, $1$, $2$, $3$, $4$이고 확률의 총합은 $1$이므로 ${\mathrm{P}} \left( X \right.$$= 0 ) + {\mathrm{P}} \left( X \right.$$= 1 ) + {\mathrm{P}} \left( X \right.$$= 2 ) + {\mathrm{P}} \left( X \right.$$= 3 ) + {\mathrm{P}} \left( X \right.$$= 4 )$$= 1$에서 $\displaystyle \frac{1}{12} + \frac{1}{12} + \frac{3}{12} + \frac{5}{12} + a = 1$ $\therefore$ $a$$\displaystyle = \frac{1}{6}$ 이때 ${\mathrm{E}} \left( X \right)$$\displaystyle = 0 \times \frac{1}{12} + 1 \times \frac{1}{12} + 2 \times \frac{3}{12} + 3 \times \frac{5}{12} + 4 \times \frac{2}{12}$ $\displaystyle = \frac{0 + 1 + 6 + 15 + 8}{12} = \frac{30}{12} = \frac{5}{2}$ $\mathrm{E} \mathit{\left( X ^{2} \right)}$$\displaystyle = 0 ^{2} \times \frac{1}{12} + 1 ^{2} \times \frac{1}{12} + 2 ^{2} \times \frac{3}{12} + 3 ^{2} \times \frac{5}{12}$$\displaystyle + 4 ^{2} \times \frac{2}{12}$ $\displaystyle = \frac{0 + 1 + 12 + 45 + 32}{12} = \frac{90}{12} = \frac{15}{2}$ 에서 ${\mathrm{V}} \left( X \right)$$= {\mathrm{E}} \left( X ^{2} \right) - \left\{ {\mathrm{E}} \left( X \right) \right\} ^{2}$ $\displaystyle = \frac{15}{2} - \left( \frac{5}{2} \right) ^{2}$$\displaystyle = \frac{30 - 25}{4}$$\displaystyle = \frac{5}{4}$ 이므로 $\displaystyle {\mathrm{V}} \left( \frac{1}{a} X \right) = 36 {\mathrm{V}} \left( X \right) = 36 \times \frac{5}{4} = 45$