题目
Problem
The continuous random variable Y has cumulative distribution function given by
F(y) = { 0, y < 0
{ y²/21, 0 ≤ y ≤ k
{ \dfrac{2}{15}\left(6y - \dfrac{y^2}{2}\right) - \dfrac{7}{5}, k < y \leqslant 6
{ 1, y > 6
where the third piece is \dfrac{2}{15}\left(6y - \dfrac{y^2}{2}\right) - \dfrac{7}{5}.
(a) Find P(Y < k/4 | Y < k)
(2)
(b) Find the value of k
(4)
(c) Use algebraic calculus to find E(Y)
(6)
(Total for Question 6 is 12 marks)
题目中文翻译
连续随机变量 Y 的累积分布函数为
F(y) = { 0, y < 0
{ y²/21, 0 ≤ y ≤ k
{ \dfrac{2}{15}\left(6y - \dfrac{y^2}{2}\right) - \dfrac{7}{5}, k < y \leqslant 6
{ 1, y > 6
第三段为 \dfrac{2}{15}\left(6y - \dfrac{y^2}{2}\right) - \dfrac{7}{5}。
(a) 求 P(Y < k/4 | Y < k)
(b) 求 k 的值
(c) 使用代数微积分求 E(Y)
（第 6 题共 12 分）