21 Lectures on Probability and Random Processes

In the second semester of the academic year 2011-2012 and for reasons unknown, I was asked to teach a course on Probability and Random Processes to second-year Informatics students. (I have been asked why it is that for a second year running, a mathematical physicist is teaching this course. I have no answer.)

At any rate, as this was the last time that the course was given, it was my first and for the foreseeable future also my last time teaching it. Nevertheless it is a topic I find fascinating and very enjoyable and I spent quite a bit of time preparing the lectures, and not wanting this effort to go to waste, I am making the lectures available here to anyone interested.

The files are the very same presentation slides I used in class. They were created with LaTeX's beamer class, with many of the diagrams drawn with PGF/TiKZ or generated with Mathematica. Most photos and some graphics I did borrow (with thanks) from the world wide web — mostly from wikipedia.

The handout mode slides are printed "4 on 1" and with flattened transitions. They are more convenient for printing and to see at a glance the contents of the lecture.

I have tried to give an impression of the topics and examples covered in each lecture, without attempting to generate a proper table of contents.

Enjoy (or endure)!

Schedule of lectures

  1. 18 January 2012 (handout mode)
    Introduction and basic notions; outcomes, events; the langugage of set theory: distributivity, De Morgan's; σ-fields
  2. 20 January 2012 (handout mode)
    Probability as relative frequency; probability measure; Bernoulli trials: fair coins and dice; uniform probability measure; inclusion-exclusion; Boole's inequality; continuity
    Birthday problem
  3. 25 January 2012 (handout mode)
    Conditional probability; multiplication rule; independence
  4. 27 January 2012 (handout mode)
    Partition rules; conditional partition rule; Bayes's rule
    Mendelian genetics
  5. 1 February 2012 (handout mode)
    Conditional independence; discrete probability distributions; distribution function
    Uniform distribution; binomial distribution; the problem of the points; Benford's distribution;
  6. 3 February 2012 (handout mode)
    Discrete random variables; probability mass function; expectation
    Waiting and the geometric distribution; Poisson distribution
  7. 8 February 2012 (handout mode)
    Functions of a discrete random variable; moment generating function; variance and standard deviation; Approximations: from binomial to Poisson
    Bortkiewicz and the Prussian cavalry fatalities
  8. 10 February 2012 (handout mode)
    Several random variables; joint probability mass functions; marginals; independence; functions of several random variables; linearity of expectation
    Bernoulli trials with a random parameter; randomised hats; coupon collector problem
  9. 15 February 2012 (handout mode)
    Expectation of a product; covariance; correlation; Markov and Chebyshev inequalities; the law of large numbers
  10. 17 February 2012 (handout mode)
    Continuous random variables; probability density functions; cumulative distribution functions; Borel sets; exponential distribution and lack of memory; expectation
    Uniform distribution; normal distribution; exponential distribution;
  11. 29 February 2012 (handout mode)
    Functions of a continuous random variable; expectation; variance; moment generating functions; standardising the normal distribution; maximum entropy
    Gamma distribution; log-normal distributions; standard error
  12. 2 March 2012 (handout mode)
    Jointly distributed continuous random variables; joint densities; marginals; joint distributions; independence; geometric probability; sums of random variables
    Buffon's needle
  13. 7 March 2012 (handout mode)
    Convolution; independent random variables; Markov, Chebyshev and Chernoff; waiting times and the exponential distribution
  14. 9 March 2012 (handout mode)
    More approximations; normal limit of binomial and Poisson distributions; sum of i.i.d. random variables; central limit theorem
    Rounding errors; astronomical measurements
  15. 14 March 2012 (handout mode)
    Stochasticity and non-determinism; Markov chains; transition matrix
    Gambler's ruin
  16. 16 March 2012 (handout mode)
    n-step transition matrix; Chapman–Kolmogorov formula; steady state and stationary distributions; regular Markov chains
    Gambler's ruin (revisited); Google PageRank
  17. 21 March 2012 (handout mode)
    Random walks; probability generating function; conditional expectation; branching processes; hitting times for random walks
    Random sums; more Gambler's ruin; Galton–Watson process and extinction of family surnames
  18. 23 March 2012 (handout mode)
    Random walks on graphs; hitting times; mean return times; independent random walks
  19. 28 March 2012 (handout mode)
    Continuous-time Markov processes; Counting process; Poisson process; inter-arrival and waiting times; uniqueness of the exponential distribution
  20. 30 March 2012 (handout mode)
    Further properties of the exponential distribution; birth and death processes; transition rates; steady-state distribution
    Single server (M/M/1) queue
  21. 4 April 2012 (handout mode)
    Steady-state probabilities for Markov processes
    Telecom circuits and Erlang's formula; Multi-server (M/M/s) queue; the sex life of the amoeba
j.m.figueroa at ed.ac.uk