Mathematics
Probability and Statistics
Probability and statistics are branches of mathematics that deal with the analysis of uncertainty and variability. Probability focuses on predicting the likelihood of future events based on data and assumptions, while statistics involves collecting, analyzing, and interpreting data to make informed decisions and predictions. These fields are widely used in various disciplines to quantify and understand uncertainty and randomness.
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12 Key excerpts on "Probability and Statistics"
- H. T. Banks, Shuhua Hu, W. Clayton Thompson(Authors)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 2 Probability and Statistics Overview The theory of Probability and Statistics is an essential mathematical tool in the formulation of inverse problems, in the development of subsequent analysis and approaches to statistical hypothesis testing and model selection criteria, and in the study of uncertainty propagation in dynamic systems. Our cover-age of these fundamental and important topics is brief and limited in scope. Indeed, we provide in this section a few definitions and basic concepts in the theory of Probability and Statistics that are essential for the understanding of estimators, confidence intervals, model selection criteria and stochastic pro-cesses. For more information on the topics in this chapter, selected references are provided at the end of these as well as subsequent chapters. 2.1 Probability and Probability Space The set of all possible outcomes in a statistical experiment is called the sample space and is denoted by Ω. Each element ω ∈ Ω is called a sample point . A collection of outcomes in which we are interested is called an event ; that is, an event is a subset of Ω. For example, consider the experiment of rolling a six-sided die. In this case, there are six possible outcomes, and the sample space can be represented as Ω = { 1 , 2 , 3 , 4 , 5 , 6 } . (2.1) An event A might be defined as A = { 1 , 5 } , (2.2) which consists of the outcomes 1 and 5. Note that we say the event A occurs if the outcome of the experiment is in the set A . Consider another experiment of tossing a coin three times. A sample point in this experiment indicates the result of each toss; for example, HHT indicates that two heads and then a tail were observed. The sample space for this experiment has eight sample points; that is, Ω = { HHH , HHT , HTH , THH , TTH , THT , HTT , TTT } . (2.3) 3
eBook - PDF- Lorena Madrigal(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
3 Probability and Statistics In this chapter we lay down the probability foundation for the rest of the book. All statistical tests rely on some aspects of probability, whether the test is working with qualitative data such as kinship titles or continuous data such as milliliters. The rules of probability are necessary for our understanding of statistical tests, but they are also necessary for our understanding of how scientific statements should be expressed, and more broadly, how science works. As I noted in chapter 1, science works with hypotheses which are submitted to tests and are either accepted or rejected but not proven. As we will see later, when we reject a hypothesis we reject it knowing full well that we could be committing an error and what the probability is that we are making an error. Therefore, the field of probability is essential to science-making because all predictions about the occurrences of events or explanations of events with hypotheses are qualified by probability statements. According to Fisher (1993), scientific inferences involving uncertainty are accompanied by the rigorous specification of the nature and extent of the uncertainty by which they are qualified. Since probability rules are fundamental to all scientific statements and to hypothesis testing, this chapter is extremely important to an understanding of statistical reasoning and hypothesis testing. Many of you were exposed to the rules of probability for the first time in Elementary School. My students have told me that they learned the rules of probability in terms of blue and red marbles and that they forgot everything else about them except how pointless the marbles seemed at the time. Marbles may not be the best example to use, and I will try to focus on anthropological examples. However, the blue and red marbles examples remind us of the broad purpose behind learning probability in a statistics class.
eBook - PDFTeaching and Learning Mathematics
A Teacher's Guide to Recent Research and Its Application
- Marilyn Nickson(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Chapter 3 Probability and Statistics Introduction In 1982 the Cockcroft Committee reported that Surprisingly few submissions which we have received have made direct reference to the teaching of statistics' (Committee of Inquiry into the Teaching of Mathematics in Schools 1982, p. 234). They also note that 'the increasing use which is being made of statistical techniques in so many fields makes it highly desirable that those who study mathematics [at a higher level] should have some understanding of Probability and Statistics' (ibid., p. 172). The dif-ficulty of achieving this is reflected in a study by Glencross (1998) reporting the development of a Statistics anxiety rating scale' which is used with students who are described as 'apprehensive' when faced with statistics sources in the social sciences and generally approach these with a negative attitude (p. 256). The importance of the curriculum providing the activities that allow children to develop these mathe-matical skills was acknowledged and since that time at least some Probability and Statistics has been included at both primary and secondary levels in the UK. This was not necessarily the case in other countries such as the US (see Shaugnessy 1992) but it is a subject which has gained momentum over the years as being one which provides a real-life context in which arithmetical and algebraic skills, as well as spatial and representational skills, can be developed. It has had different guises at primary level (much of it embedded in pictorial representation) and currently appears in the UK National Curriculum as Data Handling. The introduction of the subject means that teachers themselves have to acquire new content knowledge as well as new teaching skills to meet the demands of the cur-riculum. However, the body of research related to it is sparse and the relative newness of Probability and Statistics (known together as stochastics) in the curriculum may account for this sparseness to a large extent.
- Amit Kumar Gorai, Snehamoy Chatterjee(Authors)
- 2022(Publication Date)
- CRC Press(Publisher)
2 Basics of Probability and Statistics DOI: 10.1201/9781003200703-2 2.1 Definition of probability Probability is defined as the chances of occurrence of any event in an experiment. The sum of all the possible outcomes is called the sample space, and a subset of sample space is known as an event. If there are S exhaustive (i.e., at least one of the events must occur), mutually exclusive (i.e., only one event occurs at a time), and equally likely outcomes of a random experiment (i.e., equal chances of occurrence of each event) and r of them are favourable to the occurrence of an event A, the probability of the occurrence of the event A is given by P (A) = r S (2.1) It is sometimes expressed as ‘odds in favour of A ’ or the ‘odds against A ’. The ‘odds in favour of A ’ is defined as the ratio of occurrence of event A to the non-occurrence of event A. On the other hand, ‘odds against A ’ is defined as the ratio of non-occurrence of event A to the occurrence of. event A Odds in favour of A = P r o b a b i l i t y o f o c c u r r e n c e o f e v e n t A P r o b a b i l i t y o f non-occurrence o f e v e n t A = r / S (S − r) / S = r S − r Again, Odds against A[--=PLGO-SEPARATOR. =--]= P r o b a b i l i t y o f non-occurrence o f e v e n t A P r o b a b i l i t y o f o c c u r r e n c e o f e v e n t A = (S − r) / S r / S = S − r r The total probability of the occurrence of any event ranges from 0 to. 1. i.e., 0 ≤ P (A) ≤ 1 P (A) = 0 indicates event A is impossible, and P (A) = 1 indicates the event is certain. In the above discussion, the discrete sample space was considered. But, the probability can also be determined for a continuous sample space. For a continuous sample space, the probability of occurrence is measured as a probability density function. The probability density function of any continuous random variable gives the relative likelihood of any outcome in a specific range
eBook - PDF- Lokesh Pandey(Author)
- 2023(Publication Date)
- Arcler Press(Publisher)
MATHEMATICS, PROBABILITY, AND STATISTICS IN ENGINEERING CHAPTER6 CONTENTS 6.1. Introduction .................................................................................... 158 6.2. Organization of Text ....................................................................... 160 6.3. Probability Tables and Computer Software ...................................... 161 6.4. Probability and Random Variables .................................................. 161 6.5. Random Variables and Probability Distributions.............................. 165 6.6. Some Important Discrete Distributions............................................ 167 6.7. Some Important Continuous Distributions....................................... 169 6.8. Observed Data and Graphical Representation ................................ 173 6.9. Introduction to Statistics.................................................................. 175 6.10. Descriptive Statistics ..................................................................... 181 6.11. Enumerative Versus Analytic Studies ............................................. 182 6.12. Collecting Data............................................................................. 183 6.13. Conclusion ................................................................................... 184 References ............................................................................................. 186 The Basics of Engineering 158 Almost all engineering and applied science undergraduate programs now include at least one fundamental course on probability and statistical inference. The acknowledgement of the necessity to introduce probability theory concepts into a wide range of scientific domains today reflects some of the major shifts in science and engineering education over the last 25 years. One of the most noteworthy changes is the increased emphasis on intricacy and precision.
eBook - PDFReliability Engineering and Risk Analysis
A Practical Guide, Third Edition
- Mohammad Modarres, Mark P. Kaminskiy, Vasiliy Krivtsov(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
15 2 Basic Reliability Mathematics Review of Probability and Statistics 2.1 INTRODUCTION In this chapter, we discuss the elements of mathematical theory that are relevant to the study of the reliability of physical objects. We begin with a presentation of basic concepts of probability. Then, we briefly consider some fundamental concepts of statistics that are used in reliability analysis. 2.2 ELEMENTS OF PROBABILITY Probability is a concept that people use formally and casually every day. Weather forecasts are probabilistic in nature. People use probability in their casual conversations to show their perception of the likely occurrence or nonoccurrence of particular events. Odds are given for the outcomes of sport events and are used in gambling. The formal use of probability concepts is widespread in sci-ence, in astronomy, biology, and engineering. In this chapter, we discuss the formal application of probability theory in the field of reliability engineering. 2.2.1 S ETS AND B OOLEAN A LGEBRA To perform operations associated with probability, it is often necessary to use sets. A set is a col-lection of items or elements, each with some specific characteristics. A set that includes all items of interest is referred to as a universal set , denoted by Ω . A subset refers to a collection of items that belong to a universal set. For example, if set Ω represents the collection of all pumps in a power plant, then the collection of electrically-driven pumps is a subset E of Ω . Graphically, the relation-ship between subsets and sets can be illustrated through Venn diagrams. The Venn diagram in Figure 2.1 shows the universal set Ω by a rectangle and subsets E 1 and E 2 by circles. It can also be seen that E 2 is a subset of E 1 . The relationship between subsets E 1 and E 2 and the universal set can be symbolized by E 2 ⊂ E 1 ⊂ Ω .
eBook - PDF- Haight(Author)
- 1963(Publication Date)
- Academic Press(Publisher)
C H A P T E R 1 Probability and St a tis tics 1 .I Introduction Experiments can be classified into one of two categories, depend- ing on whether their outcomes are certain or uncertain. A certain experiment will yield exactly the same value whenever the exper- iment is repeated under the same conditions; an uncertain exper- iment will give a variety of values. The distinction is pragmatic rather than logical, for many certain experiments can be made uncertain by various methods, particularly refinement of in- strumentation. The repeated measurement of length of a line may give the same value if the ruler is coarsely calibrated and several different values if the ruler is more finely calibrated. Nevertheless, the distinction between certainty and uncertainty is a useful concept, and the mathematical form of the result is quite different. In a certain experiment, the result will consist of a single constant for each quantity measured. The result of an uncertain experiment is called a random variable; the study of uncertain results is the science of statistics. A random variable is completely described not by a number but by a functiont which shows the relative frequency (empirical or theoretical) of occurrence of partic- ular values. The function is called a statistical distribution. Consider the simple experiment of measuring the elapsed time between the arrival of the front bumpers of consecutive cars in a single lane of traffic. If the cars are rigidly scheduled, each value will be exactly the same, and the experiment will not be statistical. As we shall see there are several functions which are essentially equivalent. 1 2 Probability and Statistics However, in an actual traffic situation, many different numbers will be obtained, and these correspond to a statistical distribution.
- Simon Pollard(Author)
- 2008(Publication Date)
- IWA Publishing(Publisher)
Unit 2 Basic statistics and probability Risk management for water and wastewater utilities 15 Unit 2 requires a study time of about 4 hours. 2 Basic statistics and probability 2.1 Introduction Risk analysis requires an understanding of uncertainty and probability. Even where qualitative or semi-quantitative techniques are used, being able to recognise and apply these concepts in practice is essential. The distinction between the likelihood of an event occurring and the subsequent consequences of the event, should it occur, is the key concept in risk-based decision-making. Risk management focuses on managing either the likelihood or consequences of (usually adverse) events. Many environmental and systems reliability problems (unit 3) have to cope with sparse data and so we adopt statistical techniques to assist with these limited data sets. A note of caution however, there is always a temptation to ascribe greater power to numerical expressions of risk than can be supported by one’s knowledge of the system under study. Misplaced accuracy and precision in risk estimates may lead to misplaced confidence in systems. So, in setting out on a summary of statistics and probability theory in support of risk analysis, we need to be conscious of the limitations of these techniques, especially in light of the data they employ. Statistical methods exist for dealing with data that are either in numerical form, or can be converted to this form. In general, the data available represent a sample of some greater data set and usually shows variability; often substantive variability in orders of magnitude. For processes such as water and wastewater treatment, several data sets may be under consideration and used concurrently, each with differing levels of variability. In order to use this data in a meaningful way, information must be obtained from the collected data, in the presence of this variability, to support decisions about the underlying data population.
eBook - PDF- Timothy DelSole, Michael Tippett(Authors)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
1 Basic Concepts in Probability and Statistics Probability theory is nothing more than common sense reduced to calculation. Pierre Simon Laplace This chapter reviews some essential concepts of Probability and Statistics, including the following: • line plots, histograms, scatter plots • mean, median, quantiles, variance • random variables • probability density function • expectation of a random variable • covariance and correlation • independence • the normal distribution (also known as the Gaussian distribution) • the chi-squared distribution. These concepts provide the foundation for the statistical methods discussed in the rest of this book. 1 2 Basic Concepts in Probability and Statistics 1990 1992 1994 1996 1998 2000 25 27 29 year Nino 3.4 index (°C) Nino 3.4 index (raw) Figure 1.1 A time series of the monthly Niño 3.4 index over the period 1990–2000. 1.1 Graphical Description of Data Scientific knowledge is based on observations. However, a mere list of observational facts rarely advances science. Instead, the data need to be organized in ways that help the scientist interpret the data in a scientific interpret the data in a scientific framework and formulate new hypotheses that can be checked in independent data or experiments. To illustrate ways of describing the main characteristics of a data set, consider a specific observable quantity: the area-average sea surface temperature in the equatorial Pacific in the region 170 ◦ W − 120 ◦ W and 5 ◦ S − 5 ◦ N. This quantity is called the Niño 3.4 index and is an indicator of seasonal climate variations. The monthly average value of this index over a period of 50 or more years is readily avail- able from various data portals. What are some ways of describing such a data set? Data taken sequentially in time are known as time series. A natural way to visual- ize time series is to plot them as a function of time. A time series plot of Niño 3.4 is shown in Figure 1.1.
- Richard S. Figliola, Donald E. Beasley(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
The value of (1) can vary with repeated data sets, so the difference between it and the true average value of the population is a type of random error. Additionally, (3) requires establishing an interval within which the true average value of the population is expected to lie. This interval quantifies the probable range of this random error and is called a random uncertainty. This chapter presents an introduction to the concepts of Probability and Statistics at a level sufficient to provide information for a large class of engineering judgments. Such material allows for the reduction of raw data into results. We use terminology common to engineering, and it is not meant to replace a sound study of inferential statistics. Upon completion of this chapter, the reader will be able to: • Quantify the statistical characteristics of a data set as it relates to the population of the measured variable • Explain and use probability density functions to describe the behavior of variables • Quantify a confidence interval about the measured mean value at a given probability and to apply hypothesis tests about the behavior of engineering measurements • Perform regression analysis on a data set and quantify the confidence intervals for the param- eters of the resulting curve fit or response surface • Execute a Monte Carlo simulation that predicts the behavior expected in a result due to varia- tions in the variables involved in computing that result 97 98 Probability and Statistics 4.2 Statistical Measurement Theory Sampling refers to obtaining a set of data through repeated measurements of a variable under fixed operating conditions. This variable is known as the measured variable or, in statistical language, the measurand. We take the sampling from the population of all possible values of the measured variable.
eBook - PDFRegression Modeling
Methods, Theory, and Computation with SAS
- Michael Panik(Author)
- 2009(Publication Date)
- Chapman and Hall/CRC(Publisher)
1 1 Review of Fundamentals of Statistics Probability According to the classical view of probability , if a random experiment (e.g., tossing a fair coin a large number of times under essentially unchanged conditions) has n equiprobable out-comes, and, if n A of these outcomes constitute or favor event A , then the probability of event A , denoted P ( A ), is simply n A / n. Here P ( A ) = n A / n is termed the theoretical relative frequency of event A . Suppose P is a probability measure , a set function from the sample space S (the collection of all possible outcomes of a random experiment) to the unit interval [0,1]. Then for events A , B ⊆ S , the following probability axioms hold: A.1. P ( A ) is defined for every A ⊆ S ; A.2. P ( A ) ≥ 0 for all A ⊆ S ; A.3. P ( S ) = 1; and A.4. If A, B ⊆ S with A ∩ B = φ ( A and B are disjoint or mutually exclusive events ), then P ( A ∪ B ) = P ( A ) + P ( B ). Given A ⊆ S , a set of corollaries to the axiom system are as follows: C.1. The probability that event A does not occur is one minus the probability that it does occur or P ( ) A = 1− P ( A ); C.2. 0 ≤ P ( A ) ≤ 1; and C.3. P ( φ ) = 0. We next have the general addition rule for probabilities : for events A, B ⊆ S , P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ). (1.1) Note that, if A ∩ B = φ , then P ( A ∩ B ) = 0 and Equation 1.1 simplifies to Axiom A.4. If we are interested in the probability of occurrence of an event A given that some other event B has definitely occurred, then we compute a conditional probability , i.e., the probabil-ity of event A given B , denoted P ( A | B ), is P A B P A B P B P B ( ) ( ) ( ) , ( ) ; = ∩ ≠ 0 (1.2) 2 Regression Modeling: Methods, Theory, and Computation with SAS and the probability of B given A is P B A P A B P A P A ( ) ( ) ( ) , ( ) . = ∩ ≠ 0 (1.3) Then from Equations 1.2 and 1.3, we have the multiplication rule for probabilities or P ( A ∩ B ) = P ( A | B ) · P ( B ) = P ( B | A ) · P ( A ).
- Robert Ross(Author)
- 2018(Publication Date)
- Wiley-IEEE Press(Publisher)
25 2 Basics of Statistics and Probability The field of statistics is often divided into descriptive statistics and inferential statistics. Descriptive statistics concerns collecting and representing facts and interpretations. It usually results in an inventory to provide insight into the characteristics of a situation or population (e.g. an overview of the age of grid components, value of assets, condition of equipment, etc.). This is the type of statistics used in annual reports and performance dashboards. Descriptive statistics usually also explores correlations that may be postu- lated, investigated and tested in order to get a deeper understanding of the facts. Inferential statistics aims at decision-making based on scenarios and probabilities (e.g. deciding between replace or repair after an incident in a cable, planning a replace- ment programme, setting priorities when resources are limited or weighing the odds that a system can be energized and relied on). This requires the ability to estimate the likelihood and impact of events that may happen and to make decisions that influence the course of events. Inferential statistics aims at forecasting and what scenarios to be prepared for. Both types of statistics are used in asset management, but inferential statistics is often the biggest challenge, because it requires forecasting an unknown future and acting on it. Only the future may tell whether the decision was right, but inferences justify the decision based on the known facts and past experience. Factual data or lack thereof is generally evaluated by statistical methods. Each of the various maintenance styles described in Chapter 1 addresses its own questions on the design, operation, mainte- nance, repair and replacement of assets. The performance of each style can be evaluated statistically, whether it concerns the data, prognoses or testing of hypotheses.
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