동전으로 만든 확률변수의 평균

확률변수 $X$는 이항분포 $\displaystyle {\mathrm{B}} \mathit{\left( 4 , \frac{1}{2} \right)}$를 따르므로 $\displaystyle \mathrm{P} \mathit{\left( X = x \right)} = \mathit{_{4}} \mathrm{C} \mathit{_{\mathit{x}}} \left( \frac{1}{2} \right) ^{\mathit{4}}$ $\therefore$ $\displaystyle \mathrm{P} \mathit{\left( Y = y \right)} = \begin{cases} \mathit{_{4}} \mathrm{C} \mathit{_{\mathit{y}}} \left( \frac{1}{2} \right) ^{\mathit{4}} & & & \left( y = 0 , 1 \right) \\ 1 - \mathrm{P} \mathit{\left( Y \leq 1 \right)} & & & \left( y = 2 \right) \end{cases}$ $\therefore$ $\mathrm{E} \mathit{\left( Y \right)}$$= 0 \times \mathrm{P} \mathit{\left( Y \right.}$$= 0 ) + 1 \times \mathrm{P} \mathit{\left( Y \right.}$$= 1 ) + 2 \times \mathrm{P} \mathit{\left( Y \right.}$$= 2 )$ $\displaystyle = 0 + 1 \times \mathit{_{4}} \mathrm{C} \mathit{_{1}} \left( \frac{1}{2} \right) ^{\mathit{4}} + 2 \times \left( 1 - \mathit{_{4}} \mathrm{C} \mathit{_{0}} \left( \frac{1}{2} \right) ^{\mathit{4}} - \mathit{_{4}} \mathrm{C} \mathit{_{1}} \left( \frac{1}{2} \right) ^{\mathit{4}} \right)$ $\displaystyle = \frac{13}{8}$