### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# statistics-for-business

### View Full Document

## 9

## 0

## Popular in Course

## Popular in Business

This 150 page Document was uploaded by an elite notetaker on Friday December 18, 2015. The Document belongs to a course at a university taught by a professor in Fall. Since its upload, it has received 9 views.

## Similar to Course at University

## Reviews for statistics-for-business

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 12/18/15

Marcelo Fernandes Statistics for Business and Economics Download free books at BookBoon.com 2 Statistics for Business and Economics © 2009 Marcelo Fernandes & Ventus Publishing ApS ISBN 978-87-7681-481-6 Download free books at BookBoon.com 3 Statistics for Business and Economics Contents Contents 1. Introduction 6 1.1 Gathering data 7 1.2 Data handling 8 1.3 Probability and statistical inference 9 2. Data description 11 2.1 Data distribution 11 2.2 Typical values 13 2.3 Measures of dispersion 15 3. Basic principles of probability 18 3.1 Set theory 18 3.2 From set theory to probability 19 4. Probability distributions 36 4.1 Random variable 36 4.2 Random vectors and joint distributions 53 4.3 Marginal distributions 56 4.4 Conditional density function 57 4.5 Independent random variables 58 4.6 Expected value, moments, and co-moments 60 Please click the advert Download free books at BookBoon.com 4 Statistics for Business and Economics Contents 4.7 Discrete distributions 74 4.8 Continuous distributions 87 5. Random sampling 95 5.1 Sample statistics 99 5.2 Large-sample theory 102 6. Point and interval estimation 107 6.1 Point estimation 108 6.2 Interval estimation 121 7. Hypothesis testing 127 7.1 Rejection region for sample means 131 7.2 Size, level, and power of a test 136 7.3 Interpreting p-values 141 7.4 Likelihood-based tests 142 what‘s missing in this equation? You could be one of our future talents Please click the advert maeRsK inteRnationaL teChnoLogY & sCienCe PRogRamme Are you about to graduate as an engineer or geoscientist? Or have you already graduated? If so, there may be an exciting future for you with A.P. Moller - Maersk. www.maersk.com/mitas Download free books at BookBoon.com 5 Statistics for Business and Economics Introduction Chapter 1 Introduction This compendium aims at providing a comprehensive overview of the main topics that ap- pear in any well-structured course sequence in statistics for business and economics at the undergraduate and MBA levels. The idea is to supplement either formal or informal statistic textbooks such as, e.g., “Basic Statistical Ideas for Managers” by D.K. Hildebrand and R.L. Ott and “The Practice of Business Statistics: Using Data for Decisions” by D.S. Moore, G.P. McCabe, W.M. Duckworth and S.L. Sclove, with a summary of theory as well as with a couple of extra examples. In what follows, we set the road map for this compendium by describing the main steps of statistical analysis. www.job.oticon.dk Download free books at BookBoon.com 6 Statistics for Business and Economics Introduction Statistics is the science and art of making sense of both quantitative and qualitative data. Statistical thinking now dominates almost every ﬁeld in science, including social sciences such as business, economics, management, and marketing. It is virtually impossible to avoid data analysis if we wish to monitor and improve the quality of products and processes within a business organization. This means that economists and managers have to deal almost daily with data gathering, management, and analysis. 1.1 Gathering data Collecting data involves two key decisions. The ﬁrst refers to what to measure. Unfortu- nately, it is not necessarily the case that the easiest-to-measure variable is the most relevant for the speciﬁc problem in hand. The second relates to how to obtain the data. Sometimes gathering data is costless, e.g., a simple matter of internet downloading. However, there are many situations in which one must take a more active approach and construct a data set from scratch. Data gathering normally involves either sampling or experimentation. Albeit the latter is less common in social sciences, one should always have in mind that there is no need for a lab to run an experiment. There is pretty of room for experimentation within organizations. And we are not speaking exclusively about research and development. For instance, we could envision a sales competition to test how salespeople react to diﬀerent levels of performance incentives. This is just one example of a key driver to improve quality of products and processes. Sampling is a much more natural approach in social sciences. It is easy to appreciate that it is sometimes too costly, if not impossible, to gather universal data and hence it makes sense to restrict attention to a representative sample of the population. For instance, while census data are available only every 5 or 10 years due to the enormous cost/eﬀort that it involves, there are several household and business surveys at the annual, quarterly, monthly, and sometimes even weekly frequency. Download free books at BookBoon.com 7 Statistics for Business and Economics Introduction 1.2 Data handling Raw data are normally not very useful in that we must normally do some data manipulation before carrying out any piece of statistical analysis. Summarizing the data is the primary tool for this end. It allows us not only to assess how reliable the data are, but also to understand the main features of the data. Accordingly, it is the ﬁrst step of any sensible data analysis. Summarizing data is not only about number crunching. Actually, the ﬁrst task to trans- form numbers into valuable information is invariably to graphically represent the data. A couple of simple graphs do wonders in describing the most salient features of the data. For example, pie charts are essential to answer questions relating to proportions and fractions. For instance, the riskiness of a portfolio typically depends on how much investment there is in the risk-free asset relative to the overall investment in risky assets such as those in the equity, commodities, and bond markets. Similarly, it is paramount to map the source of problems resulting in a warranty claim so as to ensure that design and production man- agers focus their improvement eﬀorts on the right components of the product or production process. The second step is to ﬁnd the typical values of the data. It is important to know, for example, what is the average income of the households in a given residential neighborhood if you wish to open a high-end restaurant there. Averages are not suﬃcient though, for interest may sometimes lie on atypical values. It is very important to understand the probability of rare events in risk management. The insurance industry is much more concerned with extreme (rare) events than with averages. The next step is to examine the variation in the data. For instance, one of the main tenets of modern ﬁnance relates to the risk-return tradeoﬀ, where we normally gauge the riskiness of a portfolio by looking at how much the returns vary in magnitude relative to their average value. In quality control, we may improve the process by raising the average Download free books at BookBoon.com 8 Statistics for Business and Economics Introduction quality of the ﬁnal product as well as by reducing the quality variability. Understanding variability is also key to any statistical thinking in that it allows us to assess whether the variation we observe in the data is due to something other than random variation. The ﬁnal step is to assess whether there is any abnormal pattern in the data. For instance, it is interesting to examine nor only whether the data are symmetric around some value but also how likely it is to observe unusually high values that are relatively distant from the bulk of data. 1.3 Probability and statistical inference It is very diﬃcult to get data for the whole population. It is very often the case that it is too costly to gather a complete data set about a subset of characteristics in a population, either because of economic reasons or because of the computational burden. For instance, it is impossible for a ﬁrm that produces millions and millions of nails every day to check each one of their nails for quality control. This means that, in most instances, we will have to examine data coming from a sample of the population. ©2009 Accenture. All rights reserved. Always aimingfor higher ground. Just anotherday at the office for aTiger. Join the Accenture High Performance Business Forum On Thursday, April 23rd, Accenture invites top students to the High Performance Business Forum where you can learn how leading Danish companies are using the current economic downturn to gain competitive advantages. You will meet two of Accenture’s global senior executives as they present new original research and illustrate how technology can help forward Please click the adverties cope with the downturn. Visit student.accentureforum.dk to see the program and register Visit student.accentureforum.dk Download free books at BookBoon.com 9 Statistics for Business and Economics Introduction As a sample is just a glimpse of the entire population, it will entail some degree of uncer- tainty to the statistical problem. To ensure that we are able to deal with this uncertainty, it is very important to sample the data from its population in a random manner, otherwise some sort of selection bias might arise in the resulting data sample. For instance, if you wish to assess the performance of the hedge fund industry, it does not suﬃce to collect data about living hedge funds. We must also collect data on extinct funds for otherwise our database will be biased towards successful hedge funds. This sort of selection bias is also known as survivorship bias. The random nature of a sample is what makes data variability so important. Probability theory essentially aims to study how this sampling variation aﬀects statistical inference, improving our understanding how reliable our inference is. In addition, inference theory is one of the main quality-control tools in that it allows to assess whether a salient pattern in data is indeed genuine beyond reasonable random variation. For instance, some equity fund managers boast to have positive returns for a number of consecutive periods as if this would entail unrefutable evidence of genuine stock-picking ability. However, in a universe of thousands and thousands of equity funds, it is more than natural that, due to sheer luck, a few will enjoy several periods of positive returns even if the stock returns are symmetric around zero, taking positive and negative values with equal likelihood. Download free books at BookBoon.com 10 Statistics for Business and Economics Data description Chapter 2 Data description The ﬁrst step of data analysis is to summarize the data by drawing plots and charts as well as by computing some descriptive statistics. These tools essentially aim to provide a better understanding of how frequent the distinct data values are, and of how much variability there is around a typical value in the data. 2.1 Data distribution It is well known that a picture tells more than a million words. The same applies to any serious data analysis for graphs are certainly among the best and most convenient data descriptors. We start with a very simple, though extremely useful, type of data plot that reveals the frequency at which any given data value (or interval) appears in the sample. A frequency table reports the number of times that a given observation occurs or, if based on relative terms, the frequency of that value divided by the number of observations in the sample. Example A ﬁrm in the transformation industry classiﬁes the individuals at managerial positions according to their university degree. There are currently 1 accountant, 3 adminis- trators, 4 economists, 7 engineers, 2 lawyers, and 1 physicist. The corresponding frequency table is as follows. Download free books at BookBoon.com 11 Statistics for Business and Economics Data description degree accounting business economics engineering law physics value 1 2 3 4 5 6 counts 1 3 4 7 2 1 relative frequency 1/18 1/6 2/9 7/18 1/9 1/18 Note that the degree subject that a manager holds is of a qualitative nature, and so it is not particularly meaningful if one associates a number to each one of these degrees. The above table does so in the row reading ‘value’ according to the alphabetical order, for instance. The corresponding plot for this type of categorical data is the bar chart. Figure 2.1 plots a bar chart using the degrees data in the above example. This is the easiest way to identify particular shapes of the distribution of values, especially concerning data dispersion. Least data concentration occurs if the envelope of the bars forms a rectangle in that every data value appears at approximately the same frequency. it’s an interesting world Get under the skin of it. Graduate opportunities Cheltenham | £24,945 + benefits One of the UK’s intelligence services, GCHQ’s role is two-fold: to gather and analyse intelligence which helps shape Britain’s response to global events, and, to provide technical advice for the protection of Government communication and information systems. Please click the advert In doing so, our specialists – in IT, intuges,, engineering, langa information assurance, mathematics and intelligence – get well beneath the surface of global affairs. If you thought the world was an interesting place, you really ought to explore our world of work. GOVERNMENT www.careersinbritishintelligence.co.uk EMPLOYER all sections of the community. We want our workforce to reflect the diversity of our work. Download free books at BookBoon.com 12 Statistics for Business and Economics Data description In statistical quality control, one very often employs bar charts to illustrate the reasons for quality failures (in order of importance, i.e., frequency). These bar charts (also known as Pareto charts in this particular case) are indeed very popular for highlighting the natural focus points for quality improvement. Bar charts are clearly designed to describe the distribution of categorical data. In a similar vein, histograms are the easiest graphical tool for assessing the distribution of quantitative data. It is often the case that one must ﬁrst group the data into intervals before plotting a histogram. In contrast to bar charts, histogram bins are contiguous, respecting some sort of scale. 8 7 6 5 4 3 2 1 0 accounting business economics engineering law physics Figure 2.1: Bar chart of managers’ degree subjects 2.2 Typical values There are three popular measures of central tendency: mode, mean, and median. The mode refers to the most frequent observation in the sample. If a variable may take a large number of values, it is then convenient to group the data into intervals. In this instance, we deﬁne the Download free books at BookBoon.com 13 Statistics for Business and Economics Data description mode as the midpoint of the most frequent interval. Even though the mode is a very intuitive measure of central tendency, it is very sensitive to changes, even if only marginal, in data values or in the interval deﬁnition. The mean is the most commonly-used type of average and so it is often referred to simply as the average. The mean of a set of numbers is the sum ▯ of all of the elements in the set divided by the number of elements: N.=., X N Xi.f N i=1 the set is a statistical population, then we call it a population mean or expected value. If the data set is a sample of the population, we call the resulting statistic a sample mean. Finally, we deﬁne the median as the number separating the higher half of a sample/population from the lower half. We can compute the median of a ﬁnite set of numbers by sorting all the observations from lowest value to highest value and picking the middle one. Example Consider a sample of MBA graduates, whose ﬁrst salaries (in $1,000 per annum) after graduating were as follows. 75 86 86 87 89 95 95 95 95 95 96 96 96 97 97 97 97 98 98 99 99 99 99 100 100 100 105 110 110 110 115 120 122 125 132 135 140 150 150 160 165 170 172 175 185 190 200 250 250 300 The mean salary is about $126,140 per annum, whereas the median ﬁgure is exactly $100,000 and the mode amounts to $95,000. Now, if one groups the data into 8 evenly distributed bins between the minimum and maximum values, both the median and mode converge to same value of about $91,000 (i.e., the midpoint of the second bin). The mean value plays a major role in statistics. Although the median has several ad- vantages over the mean, the latter is easier to manipulate for it involves a simple linear combination of the data rather than a non-diﬀerentiable function of the data as the median. In statistical quality control, for instance, it is very common to display a means chart (also known as x-bar chart), which essentially plots the mean of a variable through time. We Download free books at BookBoon.com 14 Statistics for Business and Economics Data description say that a process is in statistical control if the means vary randomly but in a stable fash- ion, whereas it is out of statistical control if the plot shows either a dramatic variation or systematic changes. 2.3 Measures of dispersion While measures of central tendency are useful to understand what are the typical values of the data, measures of dispersion are important to describe the scatter of the data or, equivalently, data variability with respect to the central tendency. Two distinct samples may have the same mean or median, but diﬀerent levels of variability, or vice-versa. A proper description of data set should always include both of these characteristics. There are various measures of dispersion, each with its own set of advantages and disadvantages. We ﬁrst deﬁne the sample range as the diﬀerence between the largest and smallest values in the sample. This is one of the simplest measures of variability to calculate. However, it depends only on the most extreme values of the sample, and hence it is very sensitive to outliers and atypical observations. In addition, it also provides no information whatsoever about the distribution of the remaining data points. To circumvent this problem, we may think of computing the interquartile range by taking the diﬀerence between the third and ﬁrst quartiles of the distribution (i.e., subtracting the 25th percentile from the 75th percentile). This is not only a pretty good indicator of the spread in the center region of the data, but it is also much more resistant to extreme values than the sample range. We now turn our attention to the median absolute deviation, which renders a more comprehensive alternative to the interquartile range by incorporating at least partially the information from all data points in the sample. We compute the median absolute deviation bymasfmd |Xi− md(X)|, where md(·) denotes the median operator, yielding a very robust measure of dispersion to aberrant values in the sample. Finally, the most popular measure of dependence is the sample standard deviation as deﬁned by the square root of ▯ ▯ ▯ ▯ the sample variance: i.e.,Ns= 1 N Xi − XN 2,e X¯N is the sample mean. N−1 i=1 Download free books at BookBoon.com 15 Statistics for Business and Economics Data description The main advantage of variance-based measures of dispersion is that they are functions of a sample mean. In particular, the sample variance is the sample mean of the square of the deviations relative to the sample mean. Example Consider the sample of MBA graduates from the previous example. The variance of their ﬁrst salary after graduating is about $2,288,400,000 per annum, whereas the standard deviation is $47,837. The range is much larger, amounting to 300,000 − 75,000 = 225,000 per annum. The huge diﬀerence between these two measures of dispersion suggests the presence of extreme values in the data. The fact that the interquartile range is 150,000+150,00− 96,000+96,00=5 4 ,000—and hence closer the the standard deviation—seems 2 2 to corroborate this interpretation. Finally, the median absolute deviation of the sample is only 10,000 indicating that the aberrant values of the sample are among the largest (rather than smallest) values. By 2020, wind could provide one-tenth of our planet’s Brain power electricity needs. Already today, SKF’s innovative know- how is crucial to running a large proportion of the world’s wind turbines. Up to 25 % of the generating costs relate to mainte- nance. These can be reduced dramatically thanks to our systems for on-line condition monitoring and automatic lubrication. We help make it more economical to create cleaner, cheaper energy out of thin air. By sharing our experience, expertise, and creativity, industries can boost performance beyond expectations. Therefore we need the best employees who can meet this challenge! The Power of Knowledge Engineering Please click the advert Plug into The Power of Knowledge Engineering. Visit us at www.skf.com/knowledge Download free books at BookBoon.com 16 Statistics for Business and Economics Data description In statistical quality control, it is also useful to plot some measures of dispersion over time. The most common are the R and S charts, which respectively depict how the range and the standard deviation vary over time. The standard deviation is also informative in a means chart for the interval [mean value ± two standard deviations] contains about 95% of the data if their histogram is approximately bell-shaped (symmetric with a single peak). An alternative is to plot control limits at the mean value ± three standard deviations, which should include all of the data inside. These procedures are very useful in that they reduce the likelihood of a manager to go ﬁre-ﬁghting every short-term variation in the means chart. Only variations that are very likely to reﬂect something out of control will fall outside the control limits. A well-designed statistical quality-control system should take both means and dispersion charts into account for it is possible to improve on quality by reducing variability and/or by increasing average quality. For instance, a chef that reduces cooking time on average by 5 minutes, with 90% of the dishes arriving 10 minutes earlier and 10% arriving 40 minutes later, will probably not make the owner of the restaurant very happy. Download free books at BookBoon.com 17 Statistics for Business and Economics Basic principles of probability Chapter 3 Basic principles of probability 3.1 Set theory There are two fundamental sets, namely, the universe U and the empty seteyyW are fundamental because ∅⊆ A ⊆ U for every set A. Taking the diﬀerence between sets A and B yields a set whose elements are in A but not in B: A − B = {x|x ∈ A and x/∈ B}. Note that A − B is not necessarily the same as B − A. The union of A and B results in a set whose elements are in A or in B: A∪B = {x|x ∈ A or x ∈ B}. Naturally, if an element x belongs to both A and B, then it is also in the union A∪B. In turn, the intersection of A and B individuates only the elements that both sets share in common: A ∩ B = {x|x ∈ A and x ∈ B}. Last but not least, the complement A of A deﬁnes a set with all elements in the universe that are notats A,t to say, A = U − A = {x|x/∈ A}. Example Suppose that you roll a die and take note of the resulting value. The universe is the set with all possible values, namely, U = {1,2,3,4,5,6}. Consider the following two sets: A = {1,2,3,4} and B = {2,4,6}. It then follows that A − B = {1,3}, B − A = {6}, A ∪ B = {1,2,3,4,6},ad A ∩ B = {2,4}. If A and B are complementing sets, i.e., A = B, then A−B = A, B−A = B, A∪B = U, and A ∩ B = ∅. Figure 3.1 illustrates how one may represent sets using a Venn diagram. Download free books at BookBoon.com 18 Statistics for Business and Economics Basic principles of probability Figure 3.1: Venn diagram representing sets A (oval in blue and purple) and B (oval in red and purple) within the universe (rectangle box). The intersection A ∩ B of A and B is in purple, whereas the overall area in color (i.e., red, blue, and purple) corresponds to the union set A ∪ B. The complement of A consists of the areas in grey and red, whereas the areas in grey and blue deﬁne the complement of B. Properties The union and intersection operators are symmetric in that A ∪ B = B ∪ A and A ∩ B = B ∩ A. They are also transitive in that (A ∪ B) ∪ C = A ∪ (B ∪ C)a nd (A ∩ B) ∩ C = A ∩ (B ∩ C). From the above properties, it is straightforward to show that the following identities hold: (I1) A∪(B ∩C)=( A∪B)∩(A∪C), (I2) A∩(B ∪C)=( A∩B)∪(A∩C), (I3) A∩∅ = ∅, ¯ ¯ ¯ ¯ (I4) A ∪∅ = A,( ) A ∩ B = A ∪ B,( 6) A ∪ B = A ∩ B,a d(7) A = A. 3.2 From set theory to probability The probability counterpart for the universe in set theory is the sample space S. Similarly, probability focus on events, which are subsets of possible outcomes in the sample space. Example Suppose we wish to compute the probability of getting an even value in a die roll. The sample space is the universe of possible outcomes S = {1,2,3,4,5,6}, whereas the event of interest corresponds to the set {2,4,6}. Download free books at BookBoon.com 19 Statistics for Business and Economics Basic principles of probability To combine events, we employ the same rules as for sets. Accordingly, the event A ∪ B occurs if and only if we observe an outcome that belongs to A or to B,w easheent A∩B occurs if and only if both A and B happen. It is also straightforward to combine more n than two events in that ∪ i=1A iccurs if and only if at least one of the events A happensi n whereas ∩ i=1A iolds if and only if every event A occur ior i =1 ,...,n . In the same vein, the event A occurs if and only if we do not observe any outcome that belongs to the event A. Finally, we say that two events are mutually exclusive if A ∩ B = ∅, that is to say, they never occur at the same time. Mutually exclusive events are analogous to mutually exclusive sets in that their intersection is null. Trust and responsibility NNE and Pharmaplan have joined forces to create – You have to be proactive and open-minded as a NNE Pharmaplan, the world’s leading engineering newcomer and make it clear to your colleagues what and consultancy company focused entirely on the you are able to cope. The pharmaceutical ﬁeld is new pharma and biotech industries. to me. But busy as they are, most of my colleagues ﬁnd the time to teach me, and they also trust me. Inés Aréizaga Esteva (Spain), 25 years old Even though it was a bit hard at ﬁrst, I can feel over Education: Chemical Engineer time that I am beginning to be taken seriously and that my contribution is appreciated. Please click the advert NNE Pharmaplanis the world’s leading engineering and consultancy company focused entirely on the pharma and biotech industries. We employ more than our all-encompassing list of services. reannepharmaplan.comledge along with Download free books at BookBoon.com 20 Statistics for Business and Economics Basic principles of probability 3.2.1 Relative frequency Suppose we repeat a given experiment n times and count how many times, say n and A nB, the events A and B occur, respectively. It then follows that the relative frequency of event A is A = n An, whereas it is B = n Bn for event B. In addition, if events A and B are mutually exclusive (i.e., A ∩ B = ∅), then the relative frequency of C = A ∪ B is fC=( n A n )Bn = f +Af . B The relative frequency of any event is always between zero and one. Zero corresponds to an event that never occurs, whereas a relative frequency of one means that we always observe that particular event. The relative frequency is very important for the fundamental law of statistics (also known as the Glivenko-Cantelli theorem) says that, as the number of experiments n grows to inﬁnity, it converges to the probability of the eAe→ Pr(A). Chapter 5 discusses this convergence in more details. Example The Glivenko-Cantelli theorem is the principle underlying many sport compe- titions. The NBA play-oﬀs are a good example. To ensure that the team with the best odds succeed, the playoﬀs are such that a team must win a given number of games against the same adversary before qualifying to the next round. 3.2.2 Event probability It now remains to deﬁne what we exactly mean with the notion of probability. We associate a real number to the probability of observing the event A, denoted by Pr(A), satisfying the following properties: P1 0 ≤ Pr(A) ≤ 1; P2 Pr(S)=1; P3 Pr(A ∪ B)=Pr( A)+Pr( B)i f A ∩ B = ∅; Download free books at BookBoon.com 21 Statistics for Business and Economics Basic principles of probability P4 Pr(∪n A )= ▯ n Pr(A ) if the collection of events {A ,i =1 ,...,n } is pairwise i=1 i i=1 i i mutually exclusive even if n →∞ . It is easy to see that P4 follows immediately from P3 if we restrict attention to a ﬁnite number of experiments n (< ∞). From properties P1 to P4, it is possible to derive some important results concerning the diﬀerent ways we may combine events. Result It follows from P1 to P4 that (a) Pr(∅)=0, ¯ (b) Pr(A)=1 − Pr(A), (c) Pr(A ∪ B)=Pr( A)+Pr( B) − Pr(A ∩ B), and (d) Pr(A) ≤ Pr(B)i f A ⊆ B. Proof: (a) By deﬁnition, the probability of event A is the same as the probability of the union of A and ∅,vi.r( A)=Pr( A ∪∅ ). However, A and ∅ are mutually exclusive events in that A∩∅ = ∅, implying that Pr(A)=Pr( A)+Pr( ∅)by P3. (b) By deﬁnition, A∪A = S and A ∩ A = ∅, and so Pr(S)=P ( A ∪ A)=P r( A)+Pr( A)=1b y P2 and P3. (c) It ¯ ¯ is straightforward to observe that A ∪ B = A ∪ (B ∩ A)a ndht A ∩ (B ∩ A)= ∅ for the event within parentheses consists of all outcomes in B that are not in A. It thus ensues that ▯ ▯ ¯ ¯ Pr(A ∪ B)=Pr A ∪ (B ∩ A) =Pr( A)+Pr( B ∩ A). We now decompose the event B into outcomes that belong and not belong to A: B =( A∩B)∪(B ∩A). There is no intersection ¯ between these two terms, hence Pr(B)−Pr(A∩B)=Pr( B∩A), yielding the result. (d) The previous decomposition reduces to B = A ∪ (B ∩ A)g¯ intat A ∩ B = A. It then follows that Pr(B)=Pr( A)+Pr( B ∩ A) ≤ Pr(A) in view that any probability is nonnegative. ▯ 3.2.3 Finite sample space A ﬁnite sample space must have only a ﬁnite number of elements, say, {a 1a ,2..,a n}.t pjdenote the probability of observing the corresponding event {a j,ofr j =1 ,...,n .si ▯ n easy to appreciate that 0 ≤ pj≤ 1forl j =1 ,...,n and that j=1pj=1giventhatthe Download free books at BookBoon.com 22 Statistics for Business and Economics Basic principles of probability events (a ,...,a ) span the whole sample space. As the latter are also mutually exclusive, 1 n ▯ it follows that Pr(A)= p +j1.., +p jk = k p r for A = {a j1..,a k},wth1 ≤ k ≤ n. r=1 Example: The sample space corresponding to the value we obtain by throwing a die is {1,2,3,4,5,6} and the probability p ofjobserving any value j ∈{ 1,..., 6} is equal to 1/6. In general, if every element in the sample space is equiprobable, then the probability of observing a given event is equal to the ratio between the number of elements in the event and the number of elements in the sample space. Examples (1) Suppose the interest lies on the event of observing a value above 4 in a die throw. There are only two values in the sample space that satisfy this condition, namely, {5,6}, and hence the probability of this event is 2/6=1 /3. (2) Consider now ﬂipping twice a coin and recording the heads and tails. The resulting sample space is {HH,HT,TH,TT }. As the elements of the sample space are equiprobable, #{HT,TH } the probability of observing only one head is =2 /4=1 /2. #{HH,HT,TH,TT} These examples suggest that the most straightforward manner to compute the proba- bility of a given event is to run experiments in which the elements of the sample space are equiprobable. Needless to say, it is not always very easy to contrive such experiments. We illustrate this issue with another example. Example: Suppose one takes a nail from a box containing nails of three diﬀerent sizes. It is typically easier to grab a larger nail than a small one and hence such an experiment would not yield equiprobable outcomes. However, the alternative experiment in which we ﬁrst numerate the nails and then draw randomly a number to decide which nail to take would lead to equiprobable results. Download free books at BookBoon.com 23 Law for Computing Students Basic principles of probability 3.2.4 Back to the basics: Learning how to count The last example of the previous section illustrates a situation in which it is straightforward to redesign the experiment so as to induce equiprobable outcomes. Life is tough, though, and such an instance is the exception rather than the rule. For instance, a very common problem in quality control is to infer from a small random sample the probability of observing a given number of defective goods within a lot. This is evidently a situation that does not automatically lead to equiprobable outcomes given the sequential nature of the experiment. To deal with such a situation, we must ﬁrst learn how to count the possible outcomes using some tools of combinatorics. Multiplication Consider that an experiment consists of a sequence of two procedures, say, A and B.Lt n And n dBnote the number of ways in which one can execute A and B, respectively. It then follows that there is n =An B ways of executing such an experiment. In general, if the experiment consists of a sequence of k procedures, then one may run it in ▯ n = k nidiﬀerent ways. i=1 Please click the advert Download free books at BookBoon.com 24 Statistics for Business and Economics Basic principles of probability Addition Suppose now that the experiment involves k procedures in parallel (rather than in sequence). This means that we either execute the procedure 1 or the procedure 2 or ... or the procedure k.fI n ienotes the number of ways that one may carry out the procedure i ∈{ 1,...,k }, then there are n = n + ··· + n = ▯ k n ways of running such 1 k i=1 i an experiment. Permutation Suppose now that we have a set of n diﬀerent elements and we wish to know the number of sequences we can construct containing each element once, and only once. Note that the concept of sequence is distinct from that of a set, in that order of appear- ance matters. For instance, the sample space {a,b,c} allows for the following permutations ▯ n−1 (abc,acb,bac,bca,cab,cba). In general, there are n!= j=0 (n − j) possible permutations out of n elements because there are n options for the ﬁrst element of the sequence, but only n − 1 options for the second element, n − 2 options for the third element and so on until we have only one remaining option for the last element of the sequence. There is also a more general meaning for permutation in combinatorics for which we form sequences of k diﬀerent elements from a set of n elements. This means that we have n options for the ﬁrst element of the sequence, but then n − 1 options for the second element and so on until we have only n−k+1 options for the last element of the sequence. It thus follows that we have n!/(n−k)! permutations of k out of n elements in this broader sense. Combination This is a notion that only diﬀers from permutation in that ordering does not matter. This means that we just wish to know how many subsets of k elements we can construct out of a set of n elements. For instance, it is possible to form the following subsets with two elements of {a,b,c,d}: {a,b}, {a,c}, {a,d}, {b,c}, {b,d},a nd {c,d}. Note that {b,a} does not count because it is exactly the same subset as {a,b}. This suggests that, in general, the number of combinations is inferior to the number of permutations because one must count only one of the sequences that employ the same elements but with a diﬀerent ordering. In view that there are n!/(n−r)! permutations of k out of n elements and k!ways Download free books at BookBoon.com 25 Statistics for Business and Economics Basic principles of probability to choose the ordering of these k elements, the number of possible combinations of k out of n elements is ▯ ▯ n n! = . k (n − k)!k! Before we revisit the original quality control example, it is convenient to illustrate the use of the above combinatoric tools through another example. Example: Suppose there is a syndicate with 5 engineers and 3 economists. How many committees of 3 people one can form with exactly 2 engineers? Well, we must form commit- ▯ ▯ tees of 2 engineers and 1 economist. There are 5 ways of choosing 2 out of 5 engineers, 2 ▯3▯ whereas there are 1 ways of choosing 1 out of 3 economists. Altogether, this means that ▯5▯▯3▯ one can form 2 1 = 30 committees with 2 engineers and 1 economist out of a group of 5 engineers and 3 economists. We are now ready to reconsider the quality control problem of inferring the number of defective goods within a lot. Suppose, for instance, that a lot has n objects of which n d are defective and that we draw a sample of k elements of which k are ddfective. We ﬁrst note ▯ n▯ ▯nd that there are k ways of choosingk elements from a lot of n goods, whereas there are kd ways of combining k defective goods from a total of n defective goods within the lot as d d ▯ ▯ well as n−n d ways of choosing ( k − k d) elements out of the (n − n )dnon-defective goods k−kd within the lot. Accordingly, the probability of observing k dedective goods within a sample of k goods is ▯ ▯▯ ▯ k n−n d kd ▯ ▯−kd (3.1) n k if there are nddefective goods within a lot of n objects. 3.2.5 Conditional probability We denote by Pr(A|B) the probability of event A given that we have already observed event B. Intuitively, conditioning on the realization of a given event has the eﬀect of reducing the Download free books at BookBoon.com 26 Statistics for Business and Economics Basic principles of probability sample space from S to the sample space spanned by B. Examples (1) Suppose that we throw a die twice. In the ﬁrst throw, we observe a value equal to 6 and we wish to know what is the probability of observing a value of 2 in the second throw. In this instance, the fact that we have observed a value of 6 in the ﬁrst throw has no impact in the value we will observe in the second throw for the two events are independent. This means that the ﬁrst value brings about no information about the second throw and hence the probability of observing a value of 2 in the second throw given that we have observed a value of 6 in the ﬁrst throw remains the same as before, that is to say, the probability of observing a value of 2: 1/6. Please click the advert Download free books at BookBoon.com 27 Statistics for Business and Economics Basic principles of probability (2) Next, consider A = {(x ,x 1|x2+ x1 2 =01 } = {(5,5),(6,4),(4,6)} and B = {(x 1x 2|x 1 >x }2= {(2,1),(3,2),(3,1),··· ,(6,5)}. The probability of A is Pr(A)= 3/36=1 /12, whereas the probability of B is Pr(B)=1 5 /36. In addition, the probabil- ity of observing both A and B is Pr(A ∩ B)=1 /36. It thus turns out that the probability of observing A given B is Pr(A|B)=1 /15 = Pr(A ∩ B)/Pr(B), whereas the probability of observing B given A is Pr(B|A)=1 /3=Pr( A ∩ B)/Pr(A). It is obviously not by chance that, in general, Pr(A|B)=Pr(A∩B)/Pr(B), for Pr(B) > 0. By conditioning on event B we are restricting the sample space to B and hence we must consider the probability of observing both A and B and then normalize by the measure of event B. It is as if we were computing the relative frequency at which the event A occurs given the outcomes that are possible within event B. T hioinassseenfew consider unconditional events. Indeed, the unconditional probability of A is the conditional probability of A given the sample space S, i.e., Pr(A|S)=P r A ∩S )/Pr(S)=P r A). Finally, it is also interesting to note that we may decompose the probability of A ∩ B into a conditional probability and a marginal probability, namely, Pr(A ∩ B)=P r( A|B)Pr( B)= Pr(B|A)Pr( A). Example: Suppose that a computer lab has 4 new and 2 old desktops running Windows as well as 3 new and 1 old desktops running Linux. What is the probability of a student to randomly sit in front of a desktop running Windows? What is the likelihood that this particular desktop is new given that it runs Windows? Well, there are 10 computers in the lab of which 6 run Windows. This means that the answer of the ﬁrst question is 3/5, whereas Pr(new|Windows) = Pr(new ∩ Windows)/Pr(Windows) = (4/10)/(6/10) = 2/3. Figure 3.2.5 illustrates a situation in which the events A and B are mutually exclusive and hence A∩B = ∅. In this instance, the probability of both events occurring is obviously zero and so are both conditional probabilities, i.e., Pr(A∩B)=0 ⇒ Pr(A|B)=Pr( B|A)=0. In Download free books at BookBoon.com 28 Statistics for Business and Economics Basic principles of probability contrast, Figure 3.3 depicts another polar case: A ⊂ B.N w,r A ∩ B)=Pr( A), whereas Pr(A|B)=Pr( A)/Pr(B) and Pr(B|A)=1. Decomposing a joint probability into the product of a conditional probability and of a marginal probability is a very useful tool, especially if one combines it with partitions of the sample space. Let B ,1..,B kdenote a partition of the sample space S, that is to say, (a) B i B =j∅, 1 ≤ i ▯= j ≤ k k (b) ∪ i=1B i S ((rP B i > 0, 1 ≤ i ≤ k. This partition yields the decomposition A =( A ∩ B ) ∪ 1A ∩ B ) ∪ .2. ∪ (A ∩ B )o k frny event A ∈S . The nice thing about partitions is that they are mutually exclusive and hence ▯ (A∩B )∩(A∩B )= ∅ for any 1 ≤ i ▯= j ≤ k. This means that Pr(A)= k Pr(A∩B )= i j i=1 i ▯ i=1 k Pr(A|B )ir( B )i Figure 3.2: Venn diagram representing two mutually exclusive events A (oval in blue) and B (oval in red) within the sample space (rectangle box). Download free books at BookBoon.com 29 Statistics for Business and Economics Basic principles of probability Figure 3.3: Venn diagram representing events A (oval in purple) and B ( olidand purple) within the sample space (rectangle box) such that A ∩ B = A. For instance, if we deﬁne the sample space by the possible outcomes of a die throw, we may think of several distinct partitions as, for example, (a) B = {i} for i =1 ,..., 6 i (b) B 1 {1,3,5},B = {2,4,6} (c) B 1 {1,2},B = 23,4,5},B = {6}3 Example: Consider a lot of 100 frying pans of which 20 are defective. Deﬁne the events A = {ﬁrst frying pan is defective} and B = {second frying pan is defective} within a context of sequential sampling without reposition. The probability of observing event B naturally depends on whether the ﬁrst frying pan is defective or not. Now, there are only two possible outcomes in that the ﬁrst frying pan is either defective or not. This suggests a very simple partition of the sample space based on A and A, giving way to Pr(B)=P ( B|A)Pr( A)+ ¯ ¯ Pr(B|A)Pr( A). In particular, Pr(B|A)=1 9 /99 for there are only 19 defective frying pans left among the remaining 99 frying pans if A is true. Similarly, Pr(B|A)=2 0 /99, whereas 191 20 4 1 Pr(A)=1 /5 and Pr(A)=1 −Pr(A)=4 /5. We thus conclude that Pr(B)= 99 5+ 99 5 = 5 Download free books at BookBoon.com 30 Statistics for Business and Economics Basic principles of probability In some instances, we cannot observe some events, and hence we must infer whether they are true or false given the available information. For instance, if you are in a building with no windows and someone arrives completely soaked with a broken umbrella, it sounds reasonable to infer that it is raining outside even if you cannot directly observe the weather. The Bayes rule formalizes how one should conduct such an inference based on conditional probabilities: Pr(A|B )Pr( B ) Pr(B iA)= ▯ i i i =1 ,··· ,k k Pr(A|B )Pr( B ) j=1 j j where B 1,...,B k is a partition of the sample space. In the example above, we cannot observe whether it is raining, but we may partition the sample space (i.e., weather) into B = {it is raining} and B = {it is not raining}, and then calculate the probability of B given that we observe event A = {someone arrives completely soaked with a broken umbrella}. Please click the advert Download free books at BookBoon.com 31 Statistics for Business and Economics Basic principles of probability The Bayes rule has innumerable applications in business, economics and ﬁnance. For instance, imagine you are the market maker for a given stock and that there are both informed and uninformed traders in the market. In contrast to informed traders, you do not know whether news are good or bad and hence you must infer it from the trades you observe in order to adjust your bid and ask quotes accordingly. If you observe much more traders buying than selling, then you will assign a higher probability to good news. If traders are selling much more than buying, then the likelihood of bad news rises. The Bayes rule is the mechanism at which you learn whether news are good or bad by looking at trades. 3.2.6 Independent events Consider for a moment two mutually exclusive events A and B. Knowing about A gives loads of information about the likelihood of event B. In particular, if A occurs, we know for sure that event B did not occur. More formally, the conditional probability of B given that we observe A is Pr(B|A)=P r( A ∩ B)/Pr(A)=0g vntht A ∩ B = ∅ (see Figure 3.2.5). We thus conclude that A and B are dependent events given that knowing about one entails complete information about the other. Following this reasoning, it makes sense to associate independence with lack of information content. We thus say that A and B are independent events if and only if Pr(A|B)=P r A). The latter condition means that Pr(A ∩ B)= Pr(A|B)Pr( B)=P ( A)Pr( B), which in turn is equivalent to say that Pr(B|A)=P ( B) given that Pr(A ∩ B)=Pr( B|A)Pr( A) as well. Intuitively, if A and B are independent, the probability of observing A (or B) does not depend on whether B (or A has occurred) and hence conditioning on the sample space (i.e., looking at the unconditional distribution) or on the event B makes no diﬀerence. Example: Consider a lot of 10,000 pipes of which 10% comes with some sort of in- dentation. Suppose we randomly draw two pipes from the lot and deﬁne the events A = 1 {ﬁrst pipe is in perfect conditions} and A2= {second pipe is in perfect conditions}.Ifm- pling is with reposition, then events A and A are independent and so Pr(A ∩ A )= 1 2 1 2 Download free books at BookBoon.com 32 Statistics for Business and Economics Basic principles of probability Pr(A )Pr( A )=( 0 .9) =0 .81. However, if sampling is without reposition, then Pr(A ∩ 1 2 1 8,999 A 2)=Pr( A 2A 1Pr( A 1=0 .9 9,999 which is very marginally diﬀerent from 0.81. This example illustrates well a situation in which the events are not entirely independent, though assuming independence would simplify a lot the computation of the joint probability at the expenses of a very marginal cost due to the large sample. This is just to say that sometimes it pays oﬀ to assume independence between events even if we know that, in theory, they are not utterly independent. Problem set Exercise 1. Show that Pr(A ∪ B ∪ C)=Pr( A)+Pr( B)+Pr( C) − Pr(A ∩ B) − Pr(A ∩ C) − Pr(B ∩ C) +Pr( A ∩ B ∩ C). Solution We employ a similar decomposition to the one in the proof of (c)npl, (A ∪ B) ∪ C =( A ∪ B) ∪ (C ∩ A ∪ B). As the intersection is null, Pr(A ∪ B ∪ C)=Pr( A ∪ B)+Pr( C ∩ A ∪ B). We now decompose the event C into outcomes that belong and not belong to A ∪ B: ▯ ▯ ▯ ▯ C = C ∩ (A ∪ B) ∪ C ∩ A ∪ B , ▯ ▯ ▯ ▯ yielding Pr C ∩ A ∪ B =Pr( C) − Pr C ∩ (A ∪ B) .S o,eheht ▯ ▯ Pr(A ∪ B ∪ C)=Pr( A ∪ B)+Pr( C) − Pr C ∩ (A ∪ B) ▯ ▯ =Pr( A)+Pr( B)+Pr( C) − Pr(A ∩ B) − Pr C ∩ (A ∪ B) . It remains to show that the last term equals to Pr(A ∩ C)+Pr( B ∩ C) − Pr(A ∩ B ∩ C). To appreciate this, it suﬃces to see that C ∩ (A ∪ B)=( A ∩ C) ∪ (B ∩ C), which gives way Download free books at BookBoon.com 33 Statistics for Business and Economics Basic principles of probability ▯ ▯ ▯ ▯ to Pr C ∩ (A ∪ B) =P r( A ∩ C)+Pr( B ∩ C) − Pr (A ∩ C) ∩ (B ∩ C) by P3. The last term is obviously equivalent to Pr(A ∩ B ∩ C), completing the proof. ▯ Exercise 2. Consider two events A e B. Show that the probability that only one of these events occurs is Pr(A ∪ B) − Pr(A ∩ B). Solution Let C denote the event in which we observe only one event between A or B. It then consists of every possible outcome that it is in A ∪ B and not in A ∩ B.tI is straightforward to appreciate from a Venn diagram that C =( A ∪ B) − (A ∩ B)= ¯ ¯ (A ∪ B) ∩ A ∩ B =( A ∩ B) ∪ (A ∩ B). The last representation is the easiest to manipulate for it involves mutually exclusive events. In particular, it follows immediately from (c) that Pr(C)=Pr( A)+Pr( B) − 2Pr( A ∪ B)=Pr( A ∪ B) − Pr(A ∩ B). ▯ Exercise 3. There are three plants that produce a given screw: A, Bda C.nl A produces the double of screws than B and C, whose productions are at par. In addition, quality control is better at plants A and B in that only 2% of the screws they produce are defective as opposed to 4% in plant C. Suppose that we sample one screw from the warehouse that collects all screws produced by A, nda C. What is the probability that the screw is defective? What is the probability that the defective screw is from plant A? Solution: Let A = {screw comes from plant A}, B = {screw comes from plant B}, C = {screw comes from plant C},and D = {screw is defective}. Given that A’s production is twofold, it follows that Pr(A)=1 /2 and that Pr(B)=Pr(C)=1 /4. We now decompose the event D according to whether the screw comes from A, B,o r C. The latter forms a partition because if a screw comes from a given plant it cannot come from any other plant. In addition, there are only plants A,nd,aC producing this particular screw. The decomposition yields Pr(D)=Pr( D|A)Pr( A)+Pr( D|B)Pr( B)+Pr( D|C)Pr( C) 1 1 1 =0 .02 +0 .02 +0 .04 =0 .025. 2 4 4 Download free books at BookBoon.com

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over $600 per month. I LOVE StudySoup!"

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.