题目
In the summer Kylie catches a local steam train to work each day. The published arrival time for the train is 10 am.
The random variable W is the train’s actual arrival time minus the published arrival time, in minutes. When the value of W is positive, the train is late.
The cumulative distribution function F(w) is shown in the sketch below.
(a) Specify fully the probability density function f(w) of W.
(b) Write down the value of E(W)
(c) Calculate α such that P(α ≤ W ≤ 1.6) = 0.35
A day is selected at random.
(d) Calculate the probability that on this day the train arrives between 1.2 minutes late and 2.4 minutes late.
Given that on this day the train was between 1.2 minutes late and 2.4 minutes late,
(e) calculate the probability that it was more than 2 minutes late.
A random sample of 40 days is taken.
(f) Calculate the probability that for at least 10 of these days the train is between 1.2 minutes late and 2.4 minutes late.
(Total for Question 2 is 12 marks)
题目中文翻译
夏天里，Kylie 每天搭乘当地蒸汽火车上班。火车的公布到达时间是上午 10 点。
随机变量 表示火车实际到达时间减去公布到达时间，单位为分钟。若 为正，则表示火车晚点。
下图给出了累积分布函数 的草图。该分布在 到 之间为线性上升。
(a) 完整写出 的概率密度函数 。
(b) 写下 的值。
(c) 求满足 的 。
随机选取一天。
(d) 求这一天火车晚点在 1.2 分钟到 2.4 分钟之间的概率。
已知在这一天火车晚点在 1.2 分钟到 2.4 分钟之间，
(e) 求它晚点超过 2 分钟的概率。
随机抽取 40 天。
(f) 求其中至少 10 天火车晚点在 1.2 分钟到 2.4 分钟之间的概率。
（第 2 题共 12 分）