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6.041

Probabilistic Systems Analysis and Applied Probability

Undergraduate · Historical course

Fall 2010

Topics

  • Probability models and axioms
  • Conditioning and Bayes’ rule
  • Independence
  • Counting
  • Discrete random variables; probability mass functions; expectations
  • Discrete random variable examples; joint PMFs
  • Multiple discrete random variables: expectations, conditioning, independence
  • Continuous random variables
  • Multiple continuous random variables
  • Continuous Bayes rule; derived distributions
  • Derived distributions; convolution; covariance and correlation
  • Iterated expectations; sum of a random number of random variables
  • Bernoulli process
  • Poisson process - I
  • Poisson process - II
  • Markov chains - I
  • Markov chains - II
  • Markov chains - III
  • Weak law of large numbers
  • Central limit theorem
  • Bayesian statistical inference - I
  • Bayesian statistical inference - II
  • Classical statistical inference - I
  • Classical inference - II
  • Classical inference - III; course overview

Readings

  • Reading assignments are from the course textbook:
    Listed reading
  • Bertsekas, Dimitri, and John Tsitsiklis. Introduction to Probability . 2nd ed. Athena Scientific, 2008. ISBN: 978188652923.
    Listed reading
  • LEC #: 1 · TOPICS: Probability models and axioms · READINGS: Sections 1.1–1.2
    Reading assignment
  • LEC #: 2 · TOPICS: Conditioning and Bayes’ rule · READINGS: Sections 1.3–1.4
    Reading assignment
  • LEC #: 3 · TOPICS: Independence · READINGS: Section 1.5
    Reading assignment
  • LEC #: 4 · TOPICS: Counting · READINGS: Section 1.6
    Reading assignment
  • LEC #: 5 · TOPICS: Discrete random variables; probability mass functions; expectations · READINGS: Sections 2.1–2.4
    Reading assignment
  • LEC #: 6 · TOPICS: Discrete random variable examples; joint PMFs · READINGS: Sections 2.4–2.5
    Reading assignment
  • LEC #: 7 · TOPICS: Multiple discrete random variables: expectations, conditioning, independence · READINGS: Sections 2.6–2.7
    Reading assignment
  • LEC #: 8 · TOPICS: Continuous random variables · READINGS: Sections 3.1–3.3
    Reading assignment
  • LEC #: 9 · TOPICS: Multiple continuous random variables · READINGS: Sections 3.4–3.5
    Reading assignment
  • LEC #: 10 · TOPICS: Continuous Bayes rule; derived distributions · READINGS: Sections 3.6; 4.1
    Reading assignment
  • LEC #: 11 · TOPICS: Derived distributions; convolution; covariance and correlation · READINGS: Sections 4.1–4.2
    Reading assignment
  • LEC #: 12 · TOPICS: Iterated expectations; sum of a random number of random variables · READINGS: Sections 4.3; 4.5
    Reading assignment
  • LEC #: 13 · TOPICS: Bernoulli process · READINGS: Section 6.1
    Reading assignment
  • LEC #: 14 · TOPICS: Poisson process – I · READINGS: Section 6.2
    Reading assignment
  • LEC #: 15 · TOPICS: Poisson process – II · READINGS: Section 6.2
    Reading assignment
  • LEC #: 16 · TOPICS: Markov chains – I · READINGS: Sections 7.1–7.2
    Reading assignment
  • LEC #: 17 · TOPICS: Markov chains – II · READINGS: Section 7.3
    Reading assignment
  • LEC #: 18 · TOPICS: Markov chains – III · READINGS: Section 7.3
    Reading assignment
  • LEC #: 19 · TOPICS: Weak law of large numbers · READINGS: Sections 5.1–5.3
    Reading assignment
  • LEC #: 20 · TOPICS: Central limit theorem · READINGS: Section 5.4
    Reading assignment
  • LEC #: 21 · TOPICS: Bayesian statistical inference – I · READINGS: Sections 8.1–8.2
    Reading assignment
  • LEC #: 22 · TOPICS: Bayesian statistical inference – II · READINGS: Sections 8.3–8.4
    Reading assignment
  • LEC #: 23 · TOPICS: Classical statistical inference – I · READINGS: Section 9.1
    Reading assignment
  • LEC #: 24 · TOPICS: Classical inference – II · READINGS: Sections 9.1–9.4
    Reading assignment
  • LEC #: 25 · TOPICS: Classical inference – III; course overview · READINGS: Sections 9.1–9.4
    Reading assignment
  • Bertsekas, Dimitri, and John Tsitsiklis. Introduction to Probability . 2nd ed. Athena Scientific, 2008. ISBN: 9781886529236.
    Listed reading
  • Solutions to end-of-chapter problems are available: ( PDF - 1.5MB ) ↗
    Listed reading
  • Drake, Alvin. Fundamentals of Applied Probability Theory . McGraw-Hill, 1967. ISBN: 9780070178151.
    Listed reading
  • Ross, Sheldon. A First Course in Probability . 8th ed. Prentice Hall, 2009. ISBN: 9780136033134.
    Listed reading
Spring 2006 · Probabilistic Systems Analysis and Applied Probability

Topics

  • Probability Models and Axioms
  • Set Notation, Terms and Operators (include De Morgan’s), Sample Spaces, Events, Probability Axioms and Probability Laws
  • Conditioning and Bayes’ Rule
  • Conditional Probability, Multiplication Rule, Total Probability Theorem, Baye’s Rule
  • Independence
  • Introduction to Independence, Conditional Independence
  • Baye’s Theorem, Independence and Pairwise Independence
  • Counting
  • Discrete Random Variables; Probability Mass Functions; Expectations
  • Counting; Discrete Random Variables, PMFs, Expectations
  • Probability, PMF, Means, Variances, and Independence
  • Conditional Expectation; Examples
  • Conditional Expectation, Examples
  • Multiple Discrete Random Variables
  • Multiple Discrete Random Variables, PMF
  • PMF, Conditioning and Independence
  • Continuous Random Variables - I
  • Continuous Random Variables, PMF, CDF
  • Continuous Random Variables - II
  • Marginal, Conditional Densities/Expected Values/Variances
  • Expectation and Variance, CDF Function, Expectation Theorem, Baye’s Theorem
  • Continuous Random Variables and Derived Distributions
  • Random Variables, Density Functions
  • More on Continuous Random Variables, Derived Distributions, Convolution
  • Derivation of the PMF/CDF from CDF, Derivation of Distributions from Convolutions (Discrete and Continuous)
  • Transforms
  • Transforms, Properties and Uses
  • Transforms, Simple Continuous Convolution Problem
  • Iterated Expectations
  • Iterated Expectations, Random Sum of Random Variables
  • Sum of a Random Number of Random Variables
  • Expected Value and Variance
  • Iterated Expectation, Covariance/Independence with Gaussians, Random Sum of Random Variables
  • Prediction; Covariance and Correlation
  • Recitation 13
  • Weak Law of Large Numbers
  • Correlation, Estimation, Convergence in Probability
  • Signal-to-Noise Ratio, Chebyshev Inequality
  • Bernoulli Process
  • Bernoulli Process, Split Bernoulli Process
  • Poisson Process
  • Poisson Process, Concatenation of Disconnected Intervals
  • Two Instructive Drill Problems (One Bernoulli, One Poisson)
  • Poisson Process Examples
  • Competing Exponentials, Poisson Arrivals
  • Markov Chains - I
  • Markov Chain, Recurrent State
  • Poisson Process, Conditional Expectation, Markov Chain
  • Markov Chains - II
  • Steady State Probabilities, Formulating a Markov Chain Model
  • Markov Chains - III
  • Conditional Probabilities for a Birth-death Process
  • Markov Chains: Steady State Behavior and Absorption Probabilities
  • Central Limit Theorem
  • Central Limit Theorem (cont.), Strong Law of Large Numbers

Readings

  • This section contains the reading assignments from the course textbook:
    Required
  • Bertsekas, Dimitri P., and John N. Tsitsiklis. Introduction to Probability . Belmont, MA: Athena Scientific Press , June 2002. ISBN: 188652940X. ↗
    Required
  • Drake, A. Fundamentals of Applied Probability Theory . New York, NY: McGraw-Hill, 1988. ISBN: 0070178151.
    Recommended
  • Ross, S. A First Course in Probability . Upper Saddle River, NJ: Prentice Hall, 2005. ISBN: 0131856626.
    Recommended
  • Ses #: L1 · TOPICS: Probability Models and Axioms · Readings: Sections 1.1-1.2
    Reading assignment
  • Ses #: L2 · TOPICS: Conditioning and Bayes’ Rule · Readings: Sections 1.3-1.4
    Reading assignment
  • Ses #: L3 · TOPICS: Independence · Readings: Section 1.5
    Reading assignment
  • Ses #: L4 · TOPICS: Counting · Readings: Section 1.6
    Reading assignment
  • Ses #: L5 · TOPICS: Discrete Random Variables; Probability Mass Functions; Expectations · Readings: Sections 2.1-2.4
    Reading assignment
  • Ses #: L6 · TOPICS: Conditional Expectation; Examples · Readings: Sections 2.4-2.6
    Reading assignment
  • Ses #: L7 · TOPICS: Multiple Discrete Random Variables · Readings: Section 2.7
    Reading assignment
  • Ses #: L8 · TOPICS: Continuous Random Variables - I · Readings: Sections 3.1-3.3
    Reading assignment
  • Ses #: L9 · TOPICS: Continuous Random Variables - II · Readings: Sections 3.4-3.5
    Reading assignment
  • Ses #: L10 · TOPICS: Continuous Random Variables and Derived Distributions · Readings: Section 3.6
    Reading assignment
  • Ses #: L11 · TOPICS: More on Continuous Random Variables, Derived Distributions, Convolution · Readings: Section 4.2
    Reading assignment
  • Ses #: L12 · TOPICS: Transforms · Readings: Section 4.1
    Reading assignment
  • Ses #: L13 · TOPICS: Iterated Expectations · Readings: Sections 4.3
    Reading assignment
  • Ses #: L13A · TOPICS: Sum of a Random Number of Random Variables · Readings: Section 4.4
    Reading assignment
  • Ses #: L14 · TOPICS: Prediction; Covariance and Correlation · Readings: Sections 4.5-4.6
    Reading assignment
  • Ses #: L15 · TOPICS: Weak Law of Large Numbers · Readings: Sections 7.1-7.3
    Reading assignment
  • Ses #: L16 · TOPICS: Bernoulli Process · Readings: Section 5.1
    Reading assignment
  • Ses #: L17 · TOPICS: Poisson Process · Readings: Section 5.2
    Reading assignment
  • Ses #: L18 · TOPICS: Poisson Process Examples · Readings: Section 5.2
    Reading assignment
  • Ses #: L19 · TOPICS: Markov Chains - I · Readings: Sections 6.1-6.2
    Reading assignment
  • Ses #: L20 · TOPICS: Markov Chains - II · Readings: Section 6.3
    Reading assignment
  • Ses #: L21 · TOPICS: Markov Chains - III · Readings: Section 6.4
    Reading assignment
  • Ses #: L22 · TOPICS: Central Limit Theorem · Readings: Section 7.4
    Reading assignment
  • Ses #: L23 · TOPICS: Central Limit Theorem (cont.), Strong Law of Large Numbers · Readings: Section 7.5
    Reading assignment
  • The text for this course is Introduction to Probability , by Bertsekas and Tsitsiklis. Solutions to end-of-chapter problems are also available ( PDF ). We recommend that you print out these solutions. A few of these problems will be covered in recitation and tutorial. The remaining ones can be used for self-study (for best results, always try to solve a problem on your own, before reading the solution). ↗
    Listed reading
  • Bertsekas, Dimitri P., and John N. Tsitsiklis. Introduction to Probability . Belmont, MA: Athena Scientific Press , June 2002. ISBN: 188652940X. ↗
    Listed reading
  • Drake, A. Fundamentals of Applied Probability Theory . New York, NY: McGraw-Hill, 1988. ISBN: 0070178151.
    Listed reading
  • Ross, S. A First Course in Probability . Upper Saddle River, NJ: Prentice Hall, 2005. ISBN: 0131856626.
    Listed reading
Sources