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 2: Discrete Probability Distributions
示例工作表 2：离散概率分布

(based on NCT chapter 4)
（基于 NCT 第 4 章）

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, as well as practising the types of calculations the tools require you to do.
在您的问题类中，我们将重点介绍此示例表上的示例，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) before the seminar.
您应该在研讨会之前尝试所有这些示例（如果您愿意，可以分组工作）。

Q1. The possible daily demand for "The Economist" at a station news stand has the following probabilities: 
Q1电台报摊对《经济学人》的每日可能需求量有以下概率： 

The news dealer must pay £2.50p for each copy.  Each magazine sells for £5 and unsold copies are returned to the distributor for a 25p refund.   
新闻经销商必须支付 2 英镑。每本 c opy 50 便士。 每本杂志的售价为 5 英镑，未售出的副本退还给分销商，退款 25 便士。   

When 6 copies are stocked, calculate the profit (or loss) for each possible level of demand (D).   
当库存 6 份时，计算每个可能的需求水平 （D） 的利润（或损失）。

Use this information and the probability figures for D to calculate
使用此信息和 D 的概率数字进行计算

E(Profit/stocks 6).  
E（利润/股票6）。  

How many copies should he stock to maximise his E(Profit)? 
他应该备货多少份才能最大化他的 E（利润）？

Q2. The number of patients needing admission each day to a hospital intensive care unit (ICU) has a Poisson distribution with a mean of 5.3. 
问题2.每天需要入住医院重症监护病房 （ICU） 的患者人数呈泊松分布，平均值为 5.3。 

a. Why might it be reasonable to expect that daily numbers of patients would have a Poisson distribution rather than any other distribution? 
一种。为什么可以合理地预期每日患者数将具有泊松分布而不是任何其他分布？ 

b. Assuming that patients only stay in the ICU for one day, what is the probability that an 8-bedded ICU will have to turn patients away on any particular day? 
湾。 假设患者只在 ICU 住一天，那么拥有 8 个床位的 ICU 在任何特定日期不得不将患者拒之门外的概率是多少？ 

c. How many beds would the ICU require to ensure that patients, on average , were only turned away one day per year? 
c. ICU 需要多少床位才能确保患者平均每年只被拒之门外一天？ 

d. What are the practical implications of your analyses if a hospital wishes to achieve a high level of usage for its ICU beds? 
d. 如果医院希望实现其 ICU 床位的高使用率，您的分析有哪些实际意义？ 

Q3. 60 % of the chips used by a computer manufacturing company (CMC) are provided by supplier A and the rest by supplier B. The two suppliers have the following faulty chip rates: 0.3% for supplier A and 0.8% for supplier B. Consider a random batch of 20 chips (chips cannot be distinguished by supplier).
问题 3. 计算机制造公司 （CMC） 使用的 60% 的芯片由供应商 A 提供，其余由供应商 B 提供。两个供应商的缺陷芯片率如下：供应商 A 为 0.3%，供应商 B 为 0.8%。考虑随机批次的 20 个芯片（供应商无法区分芯片）。

a. What is the expected number of faulty chips? [15 marks]
一个。 预期故障芯片的数量是多少？ [15 分]

b. What is the probability of having two or more faulty chips? Justify your answer and the choice of the probability distribution. [15 marks]
b.有两个或两个以上有故障的芯片的概率是多少？证明您的答案和概率分布的选择。[15 分]

If the daily production of computer is 1000, and each computer has one chip:
如果计算机每天生产 1000 台，每台计算机都有一个芯片：

What is the probability that 8 or more computers do not pass the final quality check due to faulty chips? Suggest two probability distributions that can be used to model this situation, and compare the answers you get from using them. [20 marks]
8 台或更多计算机由于芯片故障而未通过最终质量检查的概率是多少？ 建议两个可用于对这种情况进行建模的概率分布，并比较使用它们获得的答案。[20 分]

The company, CMC, has to decide whether to implement a quality control screening of chips when delivered by the suppliers. In this case, only the chips that pass the screening will be used in the production process. Previous experience of quality control screenings reported that in 95% of the cases it was able to correctly identify faulty chips and in 3% of the cases was giving false negatives (meaning that even though the chip was working correctly the screening identified it as faulty). The screening process will be convenient only if the average number of daily faulty computers will be smaller than 3.
CMC 公司必须决定是否对供应商交付的芯片实施质量控制筛查。在这种情况下，只有通过筛选的芯片才会用于生产过程。以前的质量控制筛查经验报告称，在 95% 的情况下，它能够正确识别有缺陷的芯片，而在 3% 的情况下出现假阴性（这意味着即使芯片正常工作，筛查也会将其识别为有缺陷）。只有当每天出现故障的计算机的平均数量小于 3 台时，筛选过程才会很方便。

d. Should the company implement the quality control screening? [20 marks]
d. 公司是否应该实施质量控制筛选？[20 分]

Q4 The manager of the hospital blood bank in Lancaster receives daily requests for a rare blood type from two hospitals, one in Lancaster and one in Morecambe. Requests for this blood are known to occur at random, at different rates from the two hospitals. The request rates per day from the two hospitals are given in the table below.
Q4 兰开斯特医院血库的经理每天都会收到 来自两家医院（一家位于兰开斯特，一家位于莫克姆）的稀有血型请求。众所周知，对这种血液的要求是随机发生的，两家医院的比率不同。下表给出了两家医院每天的请求率。

What is the probability distribution of the number of requests for the blood in a 1-week period (i.e. 7 days) from Lancaster? Justify your answer.
兰开斯特在 1 周（即 7 天）内请求血液的概率分布是多少？证明你的答案是合理的。

What is the probability of exactly 4 requests from Lancaster in a 1-week period?
Lancaster 在 1 周内收到 4 个请求的概率是多少？

What is the probability of more than 4 requests from Lancaster in a 1-week period?
Lancaster 在 1 周内收到超过 4 个请求的概率是多少？

What is the probability of more than 4 requests from Lancaster occurring in exactly three out of eight 1-week periods. Justify your answer.
在 8 个 1 周期间中的 3 个期间内，来自 Lancaster 的请求超过 4 个的概率是多少。证明你的答案是合理的。

Given that requests from Lancaster and Morecambe are independent, what is the probability of a total of exactly six requests occurring in a 1-week period across both Lancaster and Morecambe?
鉴于来自 Lancaster 和 Morecambe 的请求是独立的，那么在 1 周内，Lancaster 和 Morecambe 总共恰好发生 6 个请求的概率是多少？

Q5 The suicide rate in a certain state is 1 suicide per 100,000 inhabitants per month.
Q5某个州的自杀率为每月每 100,000 名居民 1 人自杀。

(a) Find the probability that, in a city of 400,000 inhabitants within this state, there will be 8 or more suicides in a given month.
（a） 求在该州内拥有 400,000 名居民的城市中，给定月份发生 8 起或更多自杀事件的概率。

(b) What is the probability that there will be at least 2 months during the year that will have 8 or more suicides?
（b） 一年中至少有 2 个月发生 8 次或更多自杀事件的可能性有多大？

(c) Counting the present month as month number 1, what is the probability that the first month to have 8 or more suicides will be month number ? What assumptions are you making?
（c） 将当前月份算作第 1 个月，第一个月份有 8 次或更多自杀的概率是多少 ？您做了什么假设？