题目
Problem
A continuous random variable X has cumulative distribution function F(x) given by
F(x) = { 0, x < 1
{ k(ax³ - bx² + x + 4), 1 ≤ x ≤ 2
{ 1, x > 2
where a, b and k are non-zero constants.
Given that the mode of X is 1.5
(a) show that b = 3
(3)
(b) Hence show that a = 2
(1)
(c) Show that the median of X lies between 1.4 and 1.5
(4)
(Total for Question 7 is 8 marks)
题目中文翻译
连续随机变量 X 的累积分布函数 F(x) 为
F(x) = { 0, x < 1
{ k(ax³ - bx² + x + 4), 1 ≤ x ≤ 2
{ 1, x > 2
其中 a、b 和 k 是非零常数。
已知 X 的众数为 1.5
(a) 证明 b = 3
(b) 由此证明 a = 2
(c) 证明 X 的中位数在 1.4 和 1.5 之间
（第 7 题共 8 分）