题目

Problem

(i) The probability density function of a continuous random variable $X$ is given by

$f(x) = \begin{cases} \dfrac{x^3}{20} & 1 \leqslant x \leqslant 3 \\ 0 & \text{otherwise} \end{cases}$

(a) Using algebraic integration find $E(X^2)$ . You must show your working.

(3)

Given that $E(X) = 2.42$

(b) find the value of $\text{Var}(X)$ correct to 3 significant figures.

(2)

(ii) A random sample of size 10 is taken from a continuous random variable $Y$ .

(a) Find the probability that at least 7 of these values are smaller than the upper quartile of $Y$ .

(3)

(b) Find the probability that less than or equal to 5 of these values are larger than the upper quartile of $Y$ .

(2)

(Total for Question 4 is 10 marks)

题目中文翻译

(i) 连续随机变量 $X$ 的概率密度函数为

$f(x) = \begin{cases} \dfrac{x^3}{20} & 1 \leqslant x \leqslant 3 \\ 0 & \text{otherwise} \end{cases}$

(a) 使用代数积分求 $E(X^2)$ 。你必须展示你的计算过程。

已知 $E(X) = 2.42$

(b) 求 $\text{Var}(X)$ 的值，精确到 3 位有效数字。

(ii) 从连续随机变量 $Y$ 中随机抽取大小为 10 的样本。

(a) 求其中至少 7 个值小于 $Y$ 的上四分位数的概率。

(b) 求其中不超过 5 个值大于 $Y$ 的上四分位数的概率。

（第 4 题共 10 分）

IAL 2025 Jan S2 Q4