Information about Mid-term Test, April 2, 2018
==============================================

Here are a selection of topics on which I will base the in-class test 
questions for the Mid-Term on April 2, 2018. 

A separate sheet of older sample problems can be found in "SampTst07.pdf" 
in the same directory as this page, and some current sample problems are 
given at "SampleTstS18.pdf".

SYLLABUS FOR THE MID-TERM IS: CHAPTERS 1 AND 2 AND SECTIONS 4.1-4.3 OF DURRETT.

TOPICS

(1) Calculation using conditional expectations, including: 
    (a) "First-step analysis", 
    (b) Calculations of probability and moment generating functions 
for random sums such as Branching or compound Poisson processes.

(2) Definition and basic consequences of Markov Property and Strong Markov 
property for discrete-state and discrete- or continuous-time chains, including:
    (A) Chapman-Kolmogorov equation,
    (B) calculation of finite-time and limiting joint distributions for 
Markov-chain variables.

(3) Probabilities of absorption and expected time to absorption for discrete 
and continuous time chains, using first-step analysis.

(4) Classification of states in a HMC. Checking irreducibility,
aperiodicity, recurrence. Equivalent conditions for (each of) 
irreducibility, aperiodicity, recurrence and positive-recurrence.

(5) Invariant probability distributions for discrete-state chains. 
Criteria for positive recurrence including those involving $E_x(T_x)$.

(6) Calculation of long-term average quantities for Markov chains,
when expressed in terms of functions on cycles between successive
visits to some state. 

(7) Construction and interpretation of continuous-time Markov chains
using embedded chains and exponential holding times.

(8) Equivalent definitions of Poisson processes, and theorems on 
superposition, thinning and locations of known numbers of jumps.

YOU WILL NOT BE RESPONSIBLE FOR REPRODUCING PROOFS OF THEOREMS,
BUT RATHER FOR UNDERSTANDING AND USING THE RESULTS OF THOSE 
THEOREMS IN CONCRETE EXAMPLES and short strings of deductions.