Introduction to Probability

For a discussion forum, please go to the course Moodle page.

Fri 30 April at 10am: Link to live online brief Welcome meeting

Lectures

Part I: Introduction to Probability

• Lecture 1/2: Introduction, Counting, Combinatorics, Probability space, Axioms
  
Slides:
• Pre-lecture presentation style
• Pre-lecture handout style
• Post-lecture handout style
Recordings:
• Motivation, set theory revision (video recording)
• Counting (video recording)
• Combinatorics: permutations, combinations (video recording)
• Probability space, axioms, union bound (video recording)
Exercises:
• Ross: Chapter 1, Chapter 2.1-2.5
• Dekking: Chapter 2.1-2.5
• PDF file at the end of this page


• Lecture 2/3: Conditional probabilities and Bayes’ theorem
Slides:
• Pre-lecture presentation style
• Pre-lecture handout style
• Post-lecture handout style
Recordings:
• Conditional probability (video recording)
• Bayes' theorem (video recording)
• Independence (video recording)
Exercises:
• Ross: Chapter 3.1-3.5
• Dekking: Chapter 3.1-3.4
• PDF file at the end of this page

Part II: Random Variables

• Lecture 3: Random variables, probability mass function, expectation
Slides:
• Pre-lecture presentation style
• Pre-lecture handout style
• Post-lecture handout style
Recordings:
• Random variable (video recording)
• Probability mass function (video recording)
• Cumulative distribution function (video recording)
• Expectation (video recording)
Exercises:
• Ross: Chapter 4.1-4.4
• Dekking: Chapter 4.1-4.2
• PDF file at the end of this page


• Lecture 4: Expectation properties, variance, discrete distributions
Slides:
• Pre-lecture presentation style
• Pre-lecture handout style
• Post-lecture handout style
Recordings:
• Properties of expectation (video recording)
• Variance (video recording)
• Bernoulli discrete random variable, Binomial discrete random variable (video recording)
Exercises:
• Ross: Chapter 4.5-4.6
• Dekking: Chapter 7.1-7.6, 4.3, 4.5-4.6
• PDF file at the end of this page


• Lecture 5: More discrete distributions: Poisson, Geometric, Negative
Slides:
• Pre-lecture presentation style
• Pre-lecture handout style
• Post-lecture handout style
Recordings:
• Poisson discrete random variable (video recording)
• Geometric discrete random variable (video recording)
• Negative binomial discrete random variable, Hypergeometric discrete random variable (video recording)
Exercises:
• Ross: Chapter 4.7-4.8
• Dekking: Chapter 4.4-4.6
• PDF file at the end of this page


• Lecture 6: Continuous random variables
Slides:
• Pre-lecture presentation style
• Pre-lecture handout style
• Post-lecture handout style (updated!)
Recordings:
• Continuous random variables (video recording)
• Cumulative distribution functions, expectation, variance (video recording)
• Uniform random variable, Exponential random variable (video recording)
• Normal (Gaussian) random variable (video recording)
Exercises:
• Ross: Chapter 5.1-5.5
• Dekking: Chapter 5.1-5.3, 5.5, 5.7-5.8
• PDF file at the end of this page


• Lecture 7: Marginals and Joint Distributions, Independence, Covariance and Correlation
Slides:
• presentation style
• handout style
Recordings:
• Introduction to Lectures 7-12 (video recording)
• Joint Distribution, Marginal Distribution (discrete) (video recording)
• Joint Distribution, Marginal Distribution (continuous), Independence (video recording)
• Covariance, Correlation Coefficient (video recording)
Exercises:
• Ross: Chapter 6,7.4
• Dekking: Chapter 9,10
• PDF file at the end of this page

Part III: Moments and Limit Theorems

• Lectures 8 & 9: Markov, Chebyshev, Weak Law of Large Numbers, Central Limit Theorem
Slides:
• presentation style
• handout style
Recordings:
• Introduction and Warm-up Example (video recording)
• Markov's inequality, Cheyshev's inequality (video recording)
• Weak Law of Large Numbers (video recording)
• Central Limit Theorem, Illustrations (video recording)
• Applications of the Central Limit Theorem, Cauchy Distribution Example (video recording)
• Bonus Material (non-examinable): Introduction to Moments and a Proof Sketch of the Central Limit Theorem (video recording)
Exercises:
• Ross: Chapter 8
• Dekking: Chapter 13,14
• PDF file at the end of this page

Part IV: Applications and Statistics

• Lecture 10: Statistics and Estimators
Slides:
• presentation style
• handout style
• Update: on slide 19, first line: +X_n has been replaced by -X_n. Thanks to Saksham Shah for spotting this!
• Update: the explanations on slide 20 have been expanded (it is important that T(X) is not the constant zero polynomial.) Thanks to an anynomous student for spotting this
Recordings:
• Introduction (video recording)
• Empirical Distribution, Definition of Estimator (video recording)
• Examples of Estimators, Definition of Bias (video recording)
• Analysis of Bias, Estimator for Expectaton and Variance, Non-Existence of Unbiased Estimator (video recording)
• Estimating Population Size, Mean Squared Error (video recording)
• Examples of Mean Squared Error for different Estimators (video recording)
Exercises:
• Dekking: Chapter 17.1-17.2,19,20
• PDF file at the end of this page


• Lecture 11: Testing Probability Distributions (this part is non-examinable)
Slides:
• presentation style
• handout style
Recordings:
• Introduction (video recording)
• Testing Uniformity (Part One): Motivation and Problem Definition (video recording)
• Testing Uniformity (Part Two): Algorithm and Analysis (video recording)
• Conclusion (video recording)
Exercises:
• PDF file at the end of this page


• Lecture 12: Online Algorithms (Decision Making Under Uncertainty)
Slides:
• presentation style
• handout style
Update: In Slide 14 the s/(s-1)-terms have been replaced by (s-1)/s. Thanks to an anonymous student for spotting this!
Recordings:
• Introduction (video recording)
• The (Original) Secretary Problem Part One: Definition, Illustration, Basic Strategies (video recording)
• The (Original) Secretary Problem Part Two: Optimal Strategy (video recording)
• The Secretary Problem with Payoff Definition, Conclusions (video recording)
• The Odd's Algorithm, Application of Odd's Algorithm to the Original Secretary Problem (video recording)
Exercises:
• PDF file at the end of this page




Additional Exercises

Some exercises from Ross's book for lectures 1-4: PDF
Some additional exercises for lectures 1-6: PDF (updated!)
Some additional exercises for lectures 7-12: PDF (updated Lec 8-9, Q4)



Solutions to some Exercises can be found under the Supervisors Tab