Stochastics for MSc students of Electrical Engineering

Fall semester 2019
Course code: BMETE90MX55
No. of credits: 3

  • Tuesdays from 8:15 in room E407
  • Wednesdays from 12:15 in room QBF13 on even weeks (Week 2 is 18 Sept.)

  • Office hours: Tuesdays 14-15, building I, room IB115
    You can also contact me by e-mail:

    Schedule (preliminary, updated as the semester progresses):
    1Review of basic notions of probability theory
    2Probability generating functionBasic probability problems
    3Branching processes-
    4- (faculty break)Generating function, branching processes problems
    5Poisson processes-
    6Poisson processes problemsCentral limit theorem, large deviations
    7Central limit theorem, large deviations problems-
    8Midterm 1Discrete time Markov chains
    9Discrete time Markov chains problems-
    10- (university break)Continuous time Markov chains
    11Continuous time Markov chains problems-
    13Statistics problems-
    14Questions & AnswersMidterm 2

    Final mark is based on homework assignments and midterm tests.

    Homework: a total of 7 homework assignments will be issued, with a maximum of 2 points each. A total of 10 points can be obtained from the homeworks. Publication times and deadlines will be available in time.

    The two midterm tests are each worth a maximum of 45 points. You need to reach 20 points (out of 45) for each midterm test to pass.
    Date and time of the midterm tests will be announced in time.
  • Midterm 1: 29 Oct., 8:15-9:45, room E407
  • Midterm 2: 11 Dec., 12:15-13:45, room QBF13

  • Results for midterm 2 are announced on 12 Dec., 13-14, room IB115 OR if you write me an e-mail, I will send you your result.
    Basic calculators may be used during midterm. Either or both of the midterm tests may be retaken once. Retake dates:
  • Retake for Midterm 1: 17 Dec., 10:00-, room T605 (building T)
  • Retake for Midterm 2: 20 Dec., 10:00-, room T605 (building T)

  • For the retakes, no signup on Neptun or anywhere else is necessary, just come and take it.
    Those who reached 20 points in the original midterm test may come to try to improve their score, with the option not to hand it in (even at the end).
    However, if you do hand it in, your new score will replace your old score for the corresponding midterm test even if lower (except that it cannot go below 20).

    Maximal total score is 10+45+45=100. Marks based on the total score are as follows:
  • 0-39: 1
  • 40-54: 2
  • 55-69: 3
  • 70-84: 4
  • 85-100: 5

  • Materials that can (and should) be used during the semester and the midterm tests:
  • Special distributions (pdf)
  • Statistical tests (pdf)
  • Statistical tables (z, t, chi-square) (pdf)

  • Additional material (can not be used during midterm tests):
  • Probability generating functions and branching processes (pdf)

  • Problem sheets:
  • 1. Basic probability problems (pdf) (including homework problem 1), some results and solutions (pdf)
  • 2. Generating functions, branching processes problems (pdf) (including homework problem 2), some results and solutions (pdf)
  • 3. Poisson processes problems (pdf) (including homework problem 3), some results and solutions (pdf)
  • 4. Central limit theorem and large deviation problems (pdf) (including homework problem 4), some results and solutions (pdf)
  • Practice problems for midterm 1 (pdf), some results and solutions (pdf)
  • Midterm test 1A (pdf), solutions (pdf), Midterm test 1B (pdf)
  • 5. Markov chains (pdf) (including homework problem 5), some results and solutions (pdf)
  • 6. Continuous time Markov chains (pdf) (including homework problem 6, deadline is 27 Nov), some results and solutions (pdf)
  • 7. Statistics (pdf) (including homework problem 7, deadline is 11 Dec), some solutions (pdf)
  • Practice problems for midterm 2 (pdf)

  • For specific topics, I recommend further reading (see the list of books below). I recommend Kulkarni wherever applicable. In general, the other books are more detailed and more theoretical than expected; focus on definitions, main theorems and examples rather than on proofs and lemmas.
  • Basic probability: Durrett (Probability) chapters 1.1, 1.2, 1.3, 1.6, 2.1 (pages 37-38)
  • Probability generating function: Grinstead-Snell 10.1 (from subsection Ordinary Generating Function)
  • Branching processes: Grinstead-Snell 10.1
  • Poisson process: Durrett (Stochastic Processes) 2.2, Kulkarni 3
  • Law of large numbers: Durrett (Probability) 2.4
  • Central limit theorem: Durrett (Probability) 3.4.1, Berry-Esseen: Durrett 3.4.4
  • Large deviations: Durrett 2.6 (Probability)
  • Discrete time Markov chains: Durrett (Probability) 6.1-6.7, Ross 4.1-4.6, Kulkarni 2.1-2.6
  • Continuous time Markov chains: Ross 6.1-6.5, Kulkarni 4.1-4.7, 6.3

  • Recommended reading:
  • R. Durrett: Probability: Theory and Examples. 4th edition (Cambridge University Press, 2010)
  • R. Durrett: Essentials of Stochastic Processes. 3rd edition (Springer, 2016)
  • W. Feller: An Introduction to Probability Theory. Vol 1, 3rd edition (Wiley, 1968)
  • W. Feller: An Introduction to Probability Theory. Vol 2, 3rd edition (Wiley, 1971)
  • C. M. Grinstead and J. L. Snell: Introduction to Probability, 2nd ed. (AMS, 1997)
  • V. G. Kulkarni: Introduction to Modeling and Analysis of Stochastic Systems, 2nd edition (Springer, 2011)
  • Sheldon Ross: Introduction to Probability Models (Academic Press, Elsevier 2006)
  • Bhattacharyya, Johnson: Statistical principles and Methods (Wiley, 1987)
  • A. W. van der Vaart: Asymptotic Statistics (Cambridge Uniersity Press, 1998)