자연계열 (오후) 5번

(1) $\displaystyle E(X)=E(\overline X)$이므로$\displaystyle E(2X+5)=2E(X)+5=2E(\overline X)+5=2\times\frac{13}{6}+5=\frac{28}{3}$

주어진 표로부터$\displaystyle p_1+p_2+p_3=1$ ⋯⋯①$\displaystyle E(X)=\frac{13}{6}$이므로$\displaystyle p_1+2p_2+3p_3=\frac{13}{6}$ ⋯⋯②$\displaystyle \overline X=1$인 경우는 크기가 $\displaystyle 3$인 표본을 추출했을 때 모두 $\displaystyle X=1$인 경우이므로$\displaystyle P(\overline X=1)=q_1=(p_1)^3=\frac1{27}$ ⋯⋯③위 식 ①, ②, ③을 풀면$\displaystyle p_1=\frac13,p_2=\frac16,p_3=\frac12$$\displaystyle P(X=2)=\frac16$

(3) $\displaystyle V(X)=E(X^2)-\{E(X)\}^2$$\displaystyle E(X^2)=1^2\times\frac13+2^2\times\frac16+3^2\times\frac12=\frac{11}{2}$$\displaystyle \{E(X)\}^2=\left(\frac{13}{6}\right)^2=\frac{169}{36}$따라서$\displaystyle V(X)=\frac{11}{2}-\frac{169}{36}=\frac{29}{36}$

(4) $\displaystyle \begin{aligned}V(2\overline X+5)&=2^2\times V(\overline X)=2^2\times\frac{V(X)}n=\frac43V(X)\\&=\frac{29}{27}\end{aligned}$