题目
Every morning Navtej travels from home to work. Navtej leaves home at a random time between 08:00 and 08:15
• It always takes Navtej 3 minutes to walk to the bus stop
• Buses run every 15 minutes and Navtej catches the first bus that arrives
• Once Navtej has caught the bus it always takes a further 29 minutes for Navtej to reach work
The total time, T minutes, for Navtej’s journey from home to work is modelled by a continuous uniform distribution over the interval [α, β]
(a) (i) Show that α = 32
(ii) Show that β = 47
(b) State fully the probability density function for this distribution.
(c) Find the value of
(i) E(T)
(ii) Var(T)
(d) Find the probability that the time for Navtej’s journey is within 5 minutes of 35 minutes.
(Total for Question 3 is 9 marks)
题目中文翻译
每天早上纳夫特杰从家出发去上班。纳夫特杰在 08:00 到 08:15 之间的随机时间离开家
• 纳夫特杰步行到公交站总是需要3 分钟
• 公交车每15 分钟一班，纳夫特杰乘坐到达的第一辆公交车
• 纳夫特杰上公交车后，总是还需要29 分钟到达工作地点
纳夫特杰从家到工作地点的总时间 T（分钟）用区间 [α, β] 上的连续均匀分布建模
(a) (i) 证明 α = 32
(ii) 证明 β = 47
(b) 完整说明该分布的概率密度函数。
(c) 求以下的值
(i) E(T)
(ii) Var(T)
(d) 求纳夫特杰的行程时间在35 分钟的5 分钟范围内的概率。
（第 3 题共 9 分）