Random Variables and Probability
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Topic One:.
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Introduction:
Statistics is the study of the collection, organization, analysis, interpretation and presentation of data. It deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments.
Objectives:
After careful study of this chapter you should be able to do the following:
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Identify the role that statistics can play in the engineering problem-solving process
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Discuss how variability affects the data collected and used for making engineering decisions
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Explain the difference between enumerative and analytical studies
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Discuss the different methods that engineers use to collect data
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Identify the advantages that designed experiments have in comparison to other methods of collecting engineering data
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Explain the differences between mechanistic models and empirical models
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Discuss how probability and probability models are used in engineering and science
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Topic Two:
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Introduction:
Probability (or likelihood) is a measure or estimation of how likely it is that something will happen or that a statement is true. Probabilities are given a value between 0 (0% chance or will not happen) and 1 (100% chance or will happen). The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater the number of times such event is expected to happen.
These concepts have been given an axiomatic mathematical derivation in probability theory which is used widely in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence/machine learning and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.
Objectives:
After careful study of this chapter you should be able to do the following:
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Understand and describe sample spaces and events for random experiments with graphs, tables, lists, or tree diagrams
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Interpret probabilities and use probabilities of outcomes to calculate probabilities of events in discrete sample spaces
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Calculate the probabilities of joint events such as unions and intersections from the probabilities of individual events
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Interpret and calculate conditional probabilities of events
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Determine the independence of events and use independence to calculate probabilities
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Use Bayes’ theorem to calculate
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Topic Three: Discrete Random Variables and Probability Distributions
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First Exam
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Topic Four: Continuous Random Variables and Probability Distributions
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Topic Five: Joint Probability Distributions.
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Introduction:In the section on probability distributions, we looked at discrete and continuous distributions but we only focused on single random variables. Probability distributions can, however, be applied to grouped random variables which gives rise to joint probability distributions. Here we're going to focus on 2-dimensional distributions (i.e. only two random variables) but higher dimensions (more than two variables) are also possible.Since all random variables are divided into discrete and continuous random variables, we have end up having both discrete and continuous joint probability distributions.
Objectives:
After careful study of this chapter you should be able to do the following:
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Use joint probability mass functions and joint probability density functions to calculate probabilities.
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Calculate marginal and conditional probability distributions from joint probability distributions.
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Interpret and calculate covariances and correlations between random variables.
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Use the multinomial distribution to determine probabilities.
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Understand properties of a bivariate normal distribution and be able to draw contour plots for the probability density function.
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Calculate means and variances for linear combinations of random variables and calculate probabilities for linear combinations of normally distributed random variables.
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Determine the distribution of a general function of a random variable
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Topic Six: Descriptive Statistics.
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Introduction:
Descriptive statistics is the discipline of quantitatively describing the main features of a collection of information, or the quantitative description itself. Descriptive statistics are distinguished from inferential statistics (or inductive statistics), in that descriptive statistics aim to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent.
Objectives:
After careful study of this chapter you should be able to do the following:
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Compute and interpret the sample mean, sample variance, sample standard deviation, sample median, and sample range
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Explain the concepts of sample mean, sample variance, population mean, and population variance
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Construct and interpret visual data displays, including the stem-and-leaf display, the histogram, and the box plot
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Explain the concept of random sampling
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Construct and interpret normal probability plots
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Explain how to use box plots and other data displays to visually compare two or more samples of data
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Know how to use simple time series plots to visually display the important features of time oriented data
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Topic Seven: Sampling Distributions and Point Estimation of Parameters
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Introduction:
Sample data is collected on a population to draw conclusions, or make statistical inferences, about the population.Statistical methods are used to make decisions and draw conclusions about populations. This aspect of statistics is generally called statistical inference. These techniques utilize the information in a sample in drawing conclusions. This chapter begins our study of the statistical methods used in decision making.
Objectives:
After careful study of this chapter you should be able to do the following:
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Explain the general concepts of estimating the parameters of a population or a probability distribution
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Explain the important role of the normal distribution as a sampling distribution
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Understand the central limit theorem
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Explain important properties of point estimators, including bias, variance, and mean square error
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Know how to construct point estimators using the method of moments and the method of maximum likelihood
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Know how to compute and explain the precision with which a parameter is estimated
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Know how to construct a point estimator using the Bayesian approach
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Topic Eight: Statistical Intervals for a Single Sample
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Introduction:
In statistics, interval estimation is the use of sample data to calculate an interval of possible (or probable) values of an unknown population parameter, in contrast to point estimation, which is a single number. Jerzy Neyman (1937) identified interval estimation ("estimation by interval") as distinct from point estimation ("estimation by unique estimate"). In doing so, he recognized that then-recent work quoting results in the form of an estimate plus-or-minus a standard deviation indicated that interval estimation was actually the problem statisticians really had in mind.
The most prevalent forms of interval estimation are:
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confidence intervals (a frequentist method); and
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credible intervals (a Bayesian method).
Other common approaches to interval estimation, which are encompassed by statistical theory, are:
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Tolerance intervals
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Prediction intervals - used mainly in Regression Analysis
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Likelihood intervals
Objectives:
After careful study of this chapter you should be able to do the following:
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Construct confidence intervals on the mean of a normal distribution, using either the normal distribution or the t distribution method
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Construct confidence intervals on the variance and standard deviation of a normal distribution
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Construct confidence intervals on a population proportion
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Use a general method for constructing an approximate confidence interval on a parameter
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Construct prediction intervals for a future observation
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Construct a tolerance interval for a normal population
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Explain the three types of interval estimates: confidence intervals, prediction intervals, and tolerance intervals
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Topic Nine: Tests of Hypotheses for a Single Sample
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Introduction:
Hypothesis testing refers to the process of choosing between competing hypotheses about a probability distribution, based on observed data from the distribution. It is a core topic in mathematical statistics, and indeed is a fundamental part of the language of statistics.
Objectives:
After careful study of this chapter you should be able to do the following:
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Structure engineering decision-making problems as hypothesis tests
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Test hypotheses on the mean of a normal distribution using either a Z-test or a t-test procedure
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Test hypotheses on the variance or standard deviation of a normal distribution
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Test hypotheses on a population proportion
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Use the P-value approach for making decisions in hypotheses tests
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Compute power and type II error probability, and make sample size selection decisions for tests on means, variances, and proportions
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Explain and use the relationship between confidence intervals and hypothesis tests
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Use the chi-square goodness of fit test to check distributional assumptions
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Use contingency table tests
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