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1.010

Probability and Causal Inference

5-0-7 units · Undergraduate

Introduces probability and causal inference with an emphasis on understanding, quantifying, and modeling uncertainty and cause-effect relationships in an engineering context. Topics in the first half include events and their probability, the total probability and Bayes' theorems, discrete and continuous random variables and vectors, and conditional analysis. Topics in the second half include covariance, correlation, regression analysis, causality analysis, structural causal models, interventions, and hypothesis testing. Concepts illustrated through data and applications. credit cannot also be received for 1.010A or 1.010B .

Fall 2008 · Uncertainty in Engineering

Topics

  • Introduction. Events and their properties
  • Probability of events. Conditional probability, total probability theorem
  • Independence, Bayes’ theorem
  • Total probability and Bayes’ theorems
  • Discrete random variables. Bernoulli and geometric distributions
  • Binomial and Poisson distributions
  • Discrete random variables
  • Continuous random variables. Uniform and exponential distributions
  • Hazard function, distributions of mixed type and distribution mixtures
  • Continuous random variables, and hazard function
  • Discrete random vectors
  • Continuous random vectors
  • Random vectors
  • Functions of random variables; linear functions
  • Functions of random variables and vectors; monotonic and min/max functions
  • Functions of random variables
  • Functions of random vectors: sums of variables, gamma distribution
  • Expectation, second moment characterization of random variables, probabilistic moments
  • Functions of random variables and vectors
  • Second moment (SM) and first order second moment (FOSM) propagation of uncertainty for variables
  • Second moment characterization of random vectors; covariance and correlation coefficient
  • Probabilistic moments, SM and FOSM propagation of uncertainty for variables
  • SM and FOSM propagation of uncertainty for random vectors
  • Variance, covariance, correlation, SM and FOSM propagation of uncertainty for random vectors
  • Conditional SM analysis for variables
  • Conditional SM analysis for vectors
  • Normal and lognormal distributions
  • Conditional SM analysis. Important distribution models
  • Beta, extreme, and multivariate normal distributions
  • Estimation of distribution parameters: general principles
  • Estimation of distribution parameters
  • Method of moments
  • Maximum likelihood and Bayesian estimation
  • Simple and multiple linear regression
  • Pre-final review

Readings

  • Ang, Alfredo H-S., and Wilson H. Tang. Probability Concepts in Engineering: Emphasis on Applications to Civil and Environmental Engineering . 2nd ed. New York, NY: John Wiley & Sons, 2006. ISBN: 9780471720645.
    Listed reading
Fall 2003 · Computing and Data Analysis for Environmental Applications

Topics

  • Course Introduction Course Logistics Repeated Trials, Virtual Experiments, Probability, Statistics
  • Programming in MATLAB® Downloading Data, Accessing MATLAB®, MATLAB® Environment, Variables, Arrays, Scripts Plotting Data
  • Descriptive Statistics Histograms, Percentiles, Mean, Median, Variance, etc. Characterizing Streamflow Data
  • Probability Experiments, Outcomes, Sample Spaces, Events, Probability, Axioms of Probability Methods for Assigning Probabilities
  • MATLAB® Operations Internal MATLAB® Functions. Common MATLAB® Operations, Element-wise Computations, Loops Translating Equations to Programs
  • Joint Probability, Independence, Repeated Trials Joint Probability, Independent Events, Repeated Trials
  • Combinatorial Methods Counting Rules, Combinatorial Techniques for Evaluating Probabilities. Examples
  • MATLAB® Tests and Loops Relational and Logical Operations, User-defined Functions, if Tests. Virtual Experiments
  • Conditional Probability and Bayes Theorem Joint Probability, Conditional Probability, Prior & Posterior Probabilities, Bayes Theorem Engineering Applications
  • Random Variables and Probability Distributions Definition of a Random Variable Cumulative Distribution Functions, Mass and Density Functions Using Distributions to Assign Probabilities
  • Virtual Experiments
  • Expectation, Functions of a Random Variable Expectation, Population Mean and Variance Defining and Functions of a Single Random Variable Solving Derived Distribution Problems with Stochastic Simulation
  • Risk Defining and Evaluating Risk Engineering Applications
  • Some Common Probability Distributions Uniform, Exponential, Normal, and Lognormal Distributions Special Properties of Normal Random Variables Fitting Distributions to Data
  • Multivariate Probability Multiple Random Variables, Joint and Conditional Distributions, Independence, Covariance and Correlation Computing Conditional Probabilities in MATLAB®
  • Functions of Many Random Variables Derived distributions for multivariate problems, moments of linear functions of several random variables. Central Limit Theorem
  • Time Series and Central Limit Theorem
  • Populations and Samples Populations, random samples. Sample statistics, moments of the sample mean and variance.
  • Estimation Estimating Distributional Properties, Assessing Estimation Error Comparing Alternative Estimators
  • Confidence Intervals Basic Concepts, Large Sample Confidence Intervals for the Population Mean Computing Large Sample Confidence Intervals
  • Testing Hypotheses about a Single Population Formulating Hypothesis Testing Problems, Definitions Large Sample Tests of Hypotheses about a Single Population Applications Using MATLAB®
  • Testing Hypotheses about Two Populations Large Sample Tests of Hypotheses about Two Populations Controlled Experiments Applications Using MATLAB®
  • Small Samples t, chi-squared and F statistics. Small sample confidence intervals and hypothesis tests. Applications using MATLAB®.
  • Analysis of Variance (ANOVA) Testing the Significance of a Single Factor, the F Test
  • Multifactor Analysis of Variance Extension of the Single-Factor Model, Significance Testing Applications on MATLAB®
  • Examples
  • Linear Regression Objectives and Assumptions of Linear Regression, Estimating Regression Coefficients, Normal Equations Some Typical Environmental Applications
  • Analyzing Regression Results Accuracy of Regression Estimates and Predictions, Prediction Confidence Intervals, Testing Significance Continuation of Environmental Examples
  • Some Practical Applications

Readings

  • D: Devore, Jay L. Probability and Statistics for Scientists and Engineers . Duxbury Press, 2000. ISBN: 0495557447.
    Listed reading
  • E: Etter, D. Introduction to MATLAB® for Engineers and Scientists . Prentice-Hall, 1996. ISBN: 0135197031.
    Listed reading
  • LEC/REC #: L1 · TOPICS: Course Introduction Course Logistics Repeated Trials, Virtual Experiments, Probability, Statistics · READINGS: D1.1
    Reading assignment
  • LEC/REC #: R1 · TOPICS: Programming in MATLAB® Downloading Data, Accessing MATLAB®, MATLAB® Environment, Variables, Arrays, Scripts Plotting Data · READINGS: E2.1-2.6
    Reading assignment
  • LEC/REC #: L3 · TOPICS: Probability Experiments, Outcomes, Sample Spaces, Events, Probability, Axioms of Probability Methods for Assigning Probabilities · READINGS: D1.2-1.4
    Reading assignment
  • LEC/REC #: R2 · TOPICS: MATLAB® Operations Internal MATLAB® Functions. Common MATLAB® Operations, Element-wise Computations, Loops Translating Equations to Programs · READINGS: E3.1-3.4
    Reading assignment
  • LEC/REC #: L4 · TOPICS: Joint Probability, Independence, Repeated Trials Joint Probability, Independent Events, Repeated Trials · READINGS: D2.1-2.2
    Reading assignment
  • LEC/REC #: L5 · TOPICS: Combinatorial Methods Counting Rules, Combinatorial Techniques for Evaluating Probabilities. Examples · READINGS: D2.3
    Reading assignment
  • LEC/REC #: R3 · TOPICS: MATLAB® Tests and Loops Relational and Logical Operations, User-defined Functions, if Tests. Virtual Experiments · READINGS: E3.5-3.6
    Reading assignment
  • LEC/REC #: L6 · TOPICS: Conditional Probability and Bayes Theorem Joint Probability, Conditional Probability, Prior & Posterior Probabilities, Bayes Theorem Engineering Applications · READINGS: D2.4-2.5
    Reading assignment
  • LEC/REC #: L8 · TOPICS: Expectation, Functions of a Random Variable Expectation, Population Mean and Variance Defining and Functions of a Single Random Variable Solving Derived Distribution Problems with Stochastic Simulation · READINGS: D3.1-3.2, E4.1-4.2
    Reading assignment
  • LEC/REC #: L9 · TOPICS: Risk Defining and Evaluating Risk Engineering Applications · READINGS: D3.3
    Reading assignment
  • LEC/REC #: L10 · TOPICS: Some Common Probability Distributions Uniform, Exponential, Normal, and Lognormal Distributions Special Properties of Normal Random Variables Fitting Distributions to Data · READINGS: D3.4-3.6, D4.3-4.6
    Reading assignment
  • LEC/REC #: L11 · TOPICS: Multivariate Probability Multiple Random Variables, Joint and Conditional Distributions, Independence, Covariance and Correlation Computing Conditional Probabilities in MATLAB® · READINGS: D5.1-5.2
    Reading assignment
  • LEC/REC #: L13 · TOPICS: Populations and Samples Populations, random samples. Sample statistics, moments of the sample mean and variance. · READINGS: D5.3-5.5
    Reading assignment
  • LEC/REC #: L14 · TOPICS: Estimation Estimating Distributional Properties, Assessing Estimation Error Comparing Alternative Estimators · READINGS: D6.1-6.2
    Reading assignment
  • LEC/REC #: L15 · TOPICS: Confidence Intervals Basic Concepts, Large Sample Confidence Intervals for the Population Mean Computing Large Sample Confidence Intervals · READINGS: D7.1-7.3
    Reading assignment
  • LEC/REC #: L16 · TOPICS: Testing Hypotheses about a Single Population Formulating Hypothesis Testing Problems, Definitions Large Sample Tests of Hypotheses about a Single Population Applications Using MATLAB® · READINGS: D8.1-8.5
    Reading assignment
  • LEC/REC #: L17 · TOPICS: Testing Hypotheses about Two Populations Large Sample Tests of Hypotheses about Two Populations Controlled Experiments Applications Using MATLAB® · READINGS: D9.1-9.3
    Reading assignment
  • LEC/REC #: L21 · TOPICS: Multifactor Analysis of Variance Extension of the Single-Factor Model, Significance Testing Applications on MATLAB® · READINGS: D11.1-11.4
    Reading assignment
  • LEC/REC #: L22 · TOPICS: Linear Regression Objectives and Assumptions of Linear Regression, Estimating Regression Coefficients, Normal Equations Some Typical Environmental Applications · READINGS: D12.1-12.2
    Reading assignment
  • LEC/REC #: L23 · TOPICS: Analyzing Regression Results Accuracy of Regression Estimates and Predictions, Prediction Confidence Intervals, Testing Significance Continuation of Environmental Examples · READINGS: D12.3-12.4
    Reading assignment
  • The primary text is Devore, Jay L. Probability and Statistics for Scientists and Engineers. Duxbury Press, 2000. (Noted by D in the readings section.) Students not familiar with MATLAB® should also consider purchasing one of the many introductory texts on this programming package. A reasonable choice that is easy to read and moderately priced is Etter, D. Introduction to MATLAB® for Engineers and Scientists. Prentice-Hall, 1996. (Noted by E in the readings section.) ↗
    Listed reading

Prerequisites & course information

Prereq: Calculus II (GIR) U (Fall) Not offered regularly; consult department 5-0-7 units

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