题目
Problem
The continuous random variable H has cumulative distribution function F(h) where
F(h) = { 0, h ≤ 0
{ h²/48, 0 < h ≤ 4
{ h/6 - 1/3, 4 < h ≤ 5
{ (3/10)h² - (2/75)h - 2/3, 5 < h ≤ d
{ 1, h > d
where d is a constant.
(a) Show that 2d² - 45d + 250 = 0
(2)
(b) Find P(H < 1.5 | 1 < H < 4.5)
(4)
(c) Find the probability density function f(h)
You may leave the limits of h in terms of d where necessary.
(3)
(Total for Question 2 is 9 marks)
题目中文翻译
连续随机变量 H 的累积分布函数为 F(h)，其中
F(h) = { 0, h ≤ 0
{ h²/48, 0 < h ≤ 4
{ h/6 - 1/3, 4 < h ≤ 5
{ (3/10)h² - (2/75)h - 2/3, 5 < h ≤ d
{ 1, h > d
其中 d 是常数。
(a) 证明 2d² - 45d + 250 = 0
(b) 求 P(H < 1.5 | 1 < H < 4.5)
(c) 求概率密度函数 f(h)
如有必要，你可以将 h 的界限用 d 表示。
（第 2 题共 9 分）