Outline of the current topic:

Poisson and Related Processes
=============================

(1) The Poisson process (PP)
----------------------------

* definition of infinitesimal transition proba
* Proof that N(t) is Poisson using differential equations and generating function

  (1.1) Distributions associated with PP
  --------------------------------------

  * proof that inter-arrival times are exponential
  * other facts

(2) Related processes
---------------------

  (2a) The inhomogeneous PP
  -------------------------

  * Let the rate depend on t
  * Time rescaling theorem

  (2b) The birth process (BP)
  --------------------------

  * Let the rate depend on population size
  * set up forward diff equation + solve partial DE using the separation of variable method

  (2.3) The birth and death process (BDP)
  ---------------------------------------

  * set up forward diff equations -- Left as HW to find P_ij(t) for the simple BDP
  * set up differential equation to find the distribution of inter-arrival times: idea is to 
    show that a differential equation could be set up for almost any quantity of interest (see
    HW6 Q2 for example).

    (2.3.1) BDP with absorbing barriers
    -----------------------------------

    * Calculate Probability of absorption: first step analysis set-up, conditioning on T_1, the
      time of the first event/transition (conditioning via the law of iterated expectations).
    * Instead, we calculated that probability using the "jump chain": idea is to reduce the process 
      to a random walk or discrete time MC when the random times spent in the states do not matter 
      for the calculation of interest.
    * Did not cover mean time until absorption to save time. Cannot use the jump chain. Would set up a first
      step analysis conditional on T1, as show earlier. (see HW6)

(3) Limiting distributions
--------------------------
  * motivation: Hard to derive Pij(t) for most processes; easier to obtain it when t goes to infinity
  * meaning of a limiting distribution
  * How to find it: take differential equations and let t tend to infinity 
  -- to finish --


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Reading assignment/completing the class notes
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(1) As always, please review and complete your notes.

(2) Back to the calculation of the probability of absorption for the BDP with absorbing barrier 
    at 0. 
    Verify that I did not make mistakes when I derived 
    (alpha_m - alpha_1) = (alpha_1 - 1 ) * sum_0^{m-1) \rho_i
    This serves as a review on solving difference equations. (Do not hand in but alert me if you cannot
    complete the derivation.)

    To find alpha_i for all i, we considered 2 cases: sum_0^{infinity} \rho_i = infinite or finite.

    * case 1: sum_0^{infinity} \rho_i = infinite implies that alpha_1 = 1, and thus that alpha_i = 1
    for all i

    * case 2: sum_0^{infinity} \rho_i = finite

    -- In that case, we could also have alpha_1 = 1, which would also yield alpha_i = 1 for all i, which 
    I did not mention in class.
    
    -- Suppose that alpha_1 = 0; determine alpha_i for all i
       For what sort of BD process does this case happen? (specify the infinitesimal transitions of the 
       process)
       I will ask some of you to help me with that in class.


    -- Now suppose that 0< alpha_1 <1
       Show that alpha_i decreases as i increases (easy), and further that alpha_i decreases to zero (not
       sure how easy this is, if this is at all true; a proof by contradiction maybe?)
       
       Please be ready to help me with that in class.

       If indeed alpha_i tends to zero as i tends to infinity, verify that the expression for
       alpha_i I gave in class is correct.


(3) On tuesday I will finish with Poisson and related processes and continuous time MC, so it might 
be helpful to read all corresponding sections again. 
Note that you do not have to understand pages 262, 263 and 264.

THIS IS COMPLETE