题目

Problem

In this question solutions relying entirely on calculator technology are not acceptable.

The continuous random variable $X$ has a probability density function given by

$f(x) = \begin{cases} \dfrac{2}{21}x & 0 \leqslant x \leqslant k \\ \dfrac{2}{15}(6 - x) & k < x \leqslant 6 \\ 0 & \text{otherwise} \end{cases}$

(a) Show that $k = 3.5$ . You must make your method clear and show all stages of your working.

(5)

(b) Use algebraic integration to find $E(X)$ . You must show all stages of your working.

(4)

Given that $E(X^2) = \dfrac{277}{24}$

(2)

(Total for Question 5 is 11 marks)

题目中文翻译

在本题中，完全依赖计算器技术的解答不可接受。

连续随机变量 $X$ 的概率密度函数为

$f(x) = \begin{cases} \dfrac{2}{21}x & 0 \leqslant x \leqslant k \\ \dfrac{2}{15}(6 - x) & k < x \leqslant 6 \\ 0 & \text{otherwise} \end{cases}$

(a) 证明 $k = 3.5$ 。你必须清楚地展示你的方法并展示所有计算步骤。

(b) 使用代数积分求 $E(X)$ 。你必须展示所有计算步骤。

已知 $E(X^2) = \dfrac{277}{24}$

（第 5 题共 11 分）

IAL 2025 Jun S2 Q5