표본평균의 확률

$\displaystyle E ( \overline{X} ) = E ( X ) = 18$이므로 $\displaystyle E ( X ) = 10 \cdot \frac{1}{2} + 20 a + 30 \left( \frac{1}{2} - a \right)$$= 20 - 10 a$$= 18$ ∴ $\displaystyle a = \frac{1}{5}$ 크기가 $2$인 표본을 복원추출할 때, $\displaystyle \overline{X} = 20$인 경우는 $10$과 $30$, $20$과 $20$, $30$과 $10$을 추출하는 경우이므로 $\displaystyle {\mathrm{P}} ( \overline{X} = 20 )$ $= {\mathrm{P}} ( X = 10 ) \cdot {\mathrm{P}} ( X = 30 )$$+ {\mathrm{P}} ( X = 20 ) \cdot {\mathrm{P}} ( X = 20 )$ $+ {\mathrm{P}} ( X = 30 ) \cdot {\mathrm{P}} ( X = 10 )$ $\displaystyle = \frac{1}{2} \cdot \frac{3}{10} + \frac{1}{5} \cdot \frac{1}{5} + \frac{3}{10} \cdot \frac{1}{2}$ $\displaystyle = \frac{3}{20} + \frac{1}{25} + \frac{3}{20}$$\displaystyle = \frac{17}{50}$