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DEPARTMENT OF MANAGEMENT SCIENCE
管理科学系

MSc Business Analytics
理学硕士商业分析

MSCI521: Statistics and Descriptive Analytics
MSCI521:统计和描述性分析

Example Sheet 3: Continuous Probability Distributions
示例工作表 3:连续概率分布

(based on NCT chapter 5)
(基于 NCT 第 5

As noted in lectures, the textbook contains many examples which are designed to give you plenty of practice using the individual ‘tools’ contained in the chapters. You should make use of these if you find them helpful.
正如讲座中所指出的,教科书包含许多示例,旨在为您提供大量练习,使用章节中包含的各个“工具”。如果您觉得它们有帮助,您应该使用这些。

At your Problem Classes we will concentrate on the examples on this example sheet, which are designed to help you practice choosing tools, aswell as practicing the types of calculations the tools require you to do.
在您的问题类中我们将专注于此示例表上的示例,这些示例旨在帮助您练习选择工具,以及练习工具要求您进行的计算类型。

You should attempt all of these examples (working in groups if you want) and come along to your Problem Class ready to discuss your answers, or the progress that you have made tackling the questions and any problems that you are encountering.
你应该尝试所有这些例子(如果你愿意,可以分组工作),并准备好参加你的问题课,讨论你的答案,或者你在解决问题和遇到的任何问题方面取得的进展。

Q1. Daily demand in litres of blood type O+ at a major hospital is assumed to follow a Normal distribution with a mean of 20 and a variance of 25.
问题 1. 假设一家大医院的 O+ 型血的每日需求量(以升为单位)遵循 Normal 分布,平均值为 20,方差为 25。

The hospital monitors daily blood demand by categorising them accordingly to whether they fall in the top 25%, second 25%, third 25% and last 25% relative to this distribution. Find the amounts of blood that define these ranges.
医院通过根据相对于此分布是否属于前 25%、第二 25%、第三 25% 和最后 25% 进行分类来监测每日血液需求。查找定义这些范围的血液量。

Each day the hospital restocks its fridge with blood supplies up to a stock level of 25 litres. Find the probability of demand not being satisfied by the end of the day.
医院每天都会为冰箱补充血液供应,最高可达 25 升。查找在一天结束时未满足需求的概率。

How many litres should be stocked at the beginning of the day to ensure that demand is met in 99% of the days?
在一天开始时应该储存多少升,以确保在 99% 的时间内满足需求?

An additional problem in keeping high stock levels of blood is that it is perishable. Assume that blood stocked for 2 days after being acquired by the hospital are returned to a central supplier. What is the probability that all the blood in stock at the start of a day is used within the next 2 days?
保持高水平血液的另一个问题是 它容易腐烂假设医院获取后储存了 2 天的血液被退回给中央供应商。 一天开始时库存中的所有血液在接下来的 2 天内用完概率是多少

Q2. In mid-afternoon cars willing to give lifts arrive at the campus hitching point at random, at a rate of 1 every 6 minutes.
问题2.在下午三点左右,愿意提供电梯的汽车6 分钟 1 次的速度随机到达校园搭车点。

What is the probability that the gap between two cars willing to give lifts is less than 6 minutes?
两辆愿意给电梯的车之间的间隔小于 6 分钟概率是多少

What is the probability that a student arriving to find the hitching point empty will wait more than 6 minutes?
到达后发现搭便车点空着的学生等待 6 分钟以上的概率是多少?

What is the probability he/she will wait less than 3 minutes?
他/她等待少于 3 分钟的概率是多少?

What is the probability of at least one car arriving in 3 minutes?
3 分钟内至少有一辆车到达的概率是多少

What is the probability that 2 or more cars willing to give lifts will arrive in each of three consecutive 10 minute periods?
2 辆或更多愿意提供电梯的汽车在连续三个 10 分钟时段内到达的概率是多少?

Q3. There is a simple model of stock market behaviour, which is that each day the value of a stock either goes up (with probability p) or goes down (with probability (1-p)).
问题3. 股票市场行为有一个简单的模型,即股票的价值每天要么上涨(概率为 p),要么下跌(概率为 (1-p))。

What probability distribution would be suitable for a given number of days on which the value of the stock went up in a period of n days? Justify your answer and state any extra assumptions that are required.
对于股票价值在 n 天内上涨的给定天数,什么样的概率分布是合适的?证明您的答案并说明所需的任何额外假设。

Consider a continuous period of 10 days of stock market activity, for which a stock which goes up with probability p=0.6. What is the probability of:
考虑连续 10 天的股票市场活动,其中一只股票上涨的概率 p=0.6。什么是概率:

Exactly 5 days on which the value rises? Use the appropriate probability formula and show your working.
该值上升的确切 5 天?使用适当的概率公式并展示您的工作。

At least 7 days on which the value rises? You may use tables in this case.
该值上升至少 7 天?在这种情况下,您可以使用表格。

Consider a continuous period of 100 days of stock market activity for a stock which goes up with probability p=0.6. What is the probability of:
考虑一只股票连续 100 天的股票市场活动,其上涨概率为 p=0.6。什么是概率:

Exactly 50 days on which the value rises?
该值上升的确切 50 天?

At least 70 days on which the value rises?
该值至少上升 70 天?

Explain how specifically the values are calculated and why you use the specific approach.
说明具体如何计算值以及为什么使用特定方法。

Suppose that on a day when the stock goes up a fixed profit of £100 is made and when the stock goes down a fixed loss of £90 is made. What is the expected profit/loss over a 10 day period if p=0.6? What is the probability that a loss is made by the end of the 10 days?
假设在股票上涨的那一天,有 100 英镑的固定利润,当股票下跌时,有 90 英镑的固定亏损。如果 p=0.6,10 天内的预期利润/亏损是多少?在 10 天结束时亏损的概率是多少?

Suppose that on a day when the stock goes up a constant profit of £10 is made and when the stock goes down a constant loss of £9 is made. What is the expected profit/loss over a 100 day period if p=0.6? What is the probability that a loss is made by the end of the 100 days? What are the implications of your results from (d) and (e) for stock trading?
假设在股票上涨的那一天,有 10 英镑的恒定利润,当股票下跌时,有 9 英镑的持续亏损。如果 p=0.6,那么 100 天内的预期利润/亏损是多少?在 100 天结束时亏损的概率是多少?(d) 和 (e) 的结果对股票交易有什么影响?

Q4. A market-research firm has discovered that the annual earnings of people in a particular city can be approximated by a Normal distribution with a mean of £25,000 and a standard deviation of £10,000. The market-research firm is particularly interested to interview people earning between £28,000 and £40,000 as these are believed to be those most likely to buy a particular type of new car.
问题 4. 一家市场研究公司发现,特定城市人们的年收入可以用平均值为 25,000 英镑和标准差为 10,000 英镑的正态分布近似。这家市场研究公司对采访收入在 28,000 英镑到 40,000 英镑之间的人特别感兴趣,因为这些人被认为最有可能购买特定类型的新车。

In a random sample of 30 people, what is the chance that at least 10 of them will be in salary range of interest?
在 30 人的随机样本中,他们中至少有 10 人在薪资感兴趣的范围内的可能性有多大?

How big should the sample be to give a 95% chance that it will contain at least 10 people in the salary range?
样本应该有多大,才能有 95% 的机会包含至少 10 个薪资范围内的人?

Q5 The life of a certain type of automobile tire is normally distributed with mean 34,000 miles and standard deviation 4000 miles.
Q5 某种类型的汽车轮胎的使用寿命呈正态分布,平均为 34,000 英里,标准差为 4000 英里。

(a) What is the probability that such a tire lasts over 40,000 miles?
(a) 这种轮胎持续行驶超过 40,000 英里的概率是多少?

(b) What is the probability that it lasts between 30,000 and 35,000 miles?
(b) 它在 30,000 到 35,000 英里之间持续的可能性有多大?

(c) Given that it has survived 30,000 miles, what is the conditional probability that the tire survives another 10,000 miles?
(c) 假设它已经存活了 30,000 英里,那么轮胎再存活 10,000 英里的条件概率是多少?