题目
Flogcar is a car hire company that only allows cars to be hired for single days. The numbers of cars hired on different days are statistically independent. The number of cars hired on a randomly selected day from Flogcar is modelled by a Poisson distribution with mean 4
(a) Calculate the probability that exactly 6 cars are hired from Flogcar on a randomly selected day.
(b) Calculate the probability that on 2 randomly selected days, the total number of cars hired from Flogcar is between 3 and 7 inclusive.
(c) Use a suitable approximation to find the probability that on 7 randomly selected days, the total number of cars hired from Flogcar is more than 30
The probability that a car hired from Flogcar is returned in good condition is 0.97
In a randomly chosen period, Flogcar hires out 100 cars.
(d) (i) Explain why the distribution of the number of cars returned in good condition would not be well approximated by the Poisson distribution.
(ii) By considering the number of cars hired out that are not returned in good condition, use a Poisson approximation to estimate the probability that more than 95 of the cars hired out in this period are returned in good condition. Show your working clearly.
(Total for Question 1 is 14 marks)
题目中文翻译
Flogcar 是一家租车公司，只允许按天租车。不同日子租出的汽车数量彼此独立。随机选取一天，Flogcar 租出的汽车数量服从均值为 4 的泊松分布。
(a) 计算随机选取的一天恰好租出 6 辆车的概率。
(b) 计算随机选取 2 天，Flogcar 租出汽车总数在 3 到 7 辆之间（含端点）的概率。
(c) 使用合适的近似，求随机选取 7 天内，Flogcar 租出汽车总数超过 30 的概率。
一辆从 Flogcar 租出的汽车被完好归还的概率为 0.97。
在某个随机选取的时期内，Flogcar 共租出 100 辆车。
(d) (i) 解释为什么“完好归还的汽车数量”分布不适合用泊松分布近似。
(ii) 通过考虑“未完好归还的汽车数量”，使用泊松近似估计该时期内完好归还的汽车超过 95 辆的概率，并清楚写出你的过程。
（第 1 题共 14 分）