MATH2750-2009: main compulsory statements, definitions, proofs
Generally, as I mentioned earlier more than once, all definitions and all proofs from
the course are examinable. If some notion or proof is not in this list but was done in
the class, it may be asked on the exam. Also, a substantial part of the topics covered
in the class was revision. You are expected to know the basic notions from the
prerequisite courses even if they are not included in any of my lists, although, of
course, they will not be the main topics on the exam. May I repeat that all “advanced
reading” and “theory for practicals” sections are not examinable.
From the Handouts
H1.1: definition—formulae (1) and (2) p.2.
H1.5: lemma 1, Proof; Definition 1.
H2.1 & H3.9: simple algorithm, definition.
H2.2: proof of Markov property for exponential distribution.
H3.2: definition.
H3.3: lemma 1, proof; lemma 3, proof.
H3.4: transition probabilities and k-step transition probabilities, definitions
H3.5: Chapman-Kolmogorov’s equations, formulae (1), (2) and (3), with proof.
H3.6: equation (4), with proof.
H3.10: equation(s) (11).
H4: gambling, proof of equation (1).
H4.2-4.3 – solution, including uniqueness, with proof.
H4.4: proof of equation (2), solution with proof.
H5.1: definition of stationarity.
H5.2: two state MP, with proofs.
H6.1: classification, all definitions.
H7.1.1: definition of continuous time jump MP.
H7.1.2: Markov property of exponential waiting time, with proof.
H7.1.3: Poisson probability, with proof. (relates to H7.3.1)
H7.1.4: “useful techniques” – recommended.
H7.2.1: Kolmogorov’s ODEs, with proof.
H7.3.1: axioms of Poisson process. (relates to H7.1.3)
H8.1.1: definition of general MP in continuous time with intensities.
H8.1.2: Chapman-Kolmogorov’s equations (from the complete probability formula).
H8.1.3: forward Kolmogorov’s ODEs, with proof.
H8.1.4: backward Kolmogorov’s ODEs, with proof.
H8.1.5: vector/matrix forms of Kolmogorov’s ODEs.
H8.2a and 8.2: probabilities of jumps to any state, with proof.
H8.3: expectations of hitting times, with proof.
H8.4: two state MPs, with all proofs.
H8.5: birth-death processes, all formulae for stationary probabilities with proofs by
substitution for the systems M/M/1/\infty, M/M/N/\infty/1 and M/M/N/1.
H8.6: general Erlang formulae.
You have to know how to solve simple linear difference equations and simple linear
differential equations, by using characteristic equations, and how to establish linear
equation systems on probabilities and expected times such as, for example, in Q1c
and Q2c of exam 2008. It is required to know how to derive Kolmogorov’s (forward
and backward) differential equations. Understand notations M/M/N/1 et al.