2025 DSE 数学延伸部分-M1 真题 答案 详解

答案(英文)

2025 HKDSE Mathematics Extended Part (Module 1) Examination Suggested Solutions by Jacky Chan

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[ \begin{array}{|c|} \hline a & \frac{k(1)^3}{3^1} + \frac{k(3)^3}{3^3} + \frac{k(9)^3}{3^9} = 1 \ & k = \frac{27}{37} \ \hline b & E(X) = 1 \times \frac{27(1)^3}{3^1} + 3 \times \frac{27(3)^3}{3^3} + 9 \times \frac{27(9)^3}{3^9} \ & = \frac{99}{37} \ E(X^2) = 1^2 \times \frac{27(1)^3}{3^1} + 3^2 \times \frac{27(3)^3}{3^3} + 9^2 \times \frac{27(9)^3}{3^9} \ = 9 \ Var(X) = E(X^2) - [E(X)]^2 \ = 9 - \left(\frac{99}{37}\right)^2 \ = \frac{2520}{1369} \ E(\overline{X}) = E(X) \ = \frac{99}{37} \hline \end{array} ]

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