题目

Problem

The continuous random variable $X$ has cumulative distribution function given by

$F(x) = \begin{cases} 0 & x < 0 \\ \dfrac{1}{6}x(x+1) & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{cases}$

(a) Find the value of $a$ such that $P(X > a) = 0.4$

Give your answer to 3 significant figures.

(3)

(b) Use calculus to find

(i) $E(X)$

(ii) $\text{Var}(X)$

(Solutions relying on calculator technology are not acceptable.)

(8)

(Total for Question 3 is 11 marks)

题目中文翻译

连续随机变量 $X$ 的累积分布函数为

$F(x) = \begin{cases} 0 & x < 0 \\ \dfrac{1}{6}x(x+1) & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{cases}$

(a) 求 $a$ 的值，使得 $P(X > a) = 0.4$

答案保留 3 位有效数字。

(b) 使用微积分求

(i) $E(X)$

(ii) $\text{Var}(X)$

（不接受依赖计算器技术的解法。）

（第 3 题共 11 分）

IAL 2026 Jan S2 A Q3