* Housekeeping Details

- Lec: Mon/Wed 2:50-4:05
- OH:  Dunno yet
- HW:  Approx 6 probs/wk; reviewed but not graded in detail
- Txt: New one; comments welcome.
- Sty: Read the book, do the problems, ask questions.  My goal
       is not to spoon-feed the book, but rather to add perspective,
       illustrate and illuminate ideas, offer examples, and help show
       how the ideas and tools are useful in the theory and application
       of (especially Bayesian) statistics.

1. Sets and Events

   Motivation:

       Most students will have taken an undergraduate calculus-based
       course in probability theory (Duke's MTH135=STA104 or STA213);
       such a course teaches about discrete and continuous random vbls
       and their distributions, joint distributions of 2 or 3 RV's, a
       little about conditional prob's and dist'ns.  Most things are
       done twice: once for discrete rv's (binomial, geometric, poisson)
       and once for continuous (uniform, normal, exponential).

       This course builds a single coherent (beautiful) structure for
       one, two, or infinitely many random variables that are discrete
       or continuous or neither, and is especially concerned with limits
       of random variables (we will see there are many sorts of limits
       to consider) and with conditional distributions, when there may
       be many (even infinitely-many) others.

       A recurring theme is application within Bayesian statistics---
       which we may view as simply probability theory on a grand scale,
       building a joint probability model for all the things we don't
       know (for example, the probability p of success in a clinical
       trial of an experimental drug) or haven't yet observed (for
       example, the number X of successes in the trial of N subjects);
       the object is usually to deduce more about the CONDITIONAL
       DISTRIBUTION of the things we care about, given the things we
       observed... like         P[ p > 0.75  |  X=8, N=10 ]


   Notation and Basic Mathematical Set-Up:       

       \Omega:  Set of possible outcomes of some "experiment"
       \omega:  One of the outcomes in \Omega

                [Idea: nature or fate chooses an \omega from \Omega;
                       alas she doesn't tell us about it]

       A, B, C: Subsets of \Omega; A is "true" if nature's \omega\in A.
       2^\Omega: All subsets of \Omega ("Power set", sometimes denoted
                with a spikey P(\Omega))

       P[]:     Probability assignment of numbers 0<= P[A] <=1 to SOME
                (maybe not all) subsets A of \Omega...

       \cal{A}: Certain collections of sets.

       X,Y,Z:   Random variables, functions X:\Omega -> E (maybe R or R^n)
       
   Operations:

        Complement;
        Union over arbitrary index set;

        Intersection over arbitrary index set; Set difference; symmetric difference;


   Relations:

        containment; equality; 

   De Morgan's Law

   Indicator Functions: 1_A

   Lim Inf:  union of intersections = EA
   Lim Sup:  intersection of unions = IO

   Example:
 

       {w: Xn(w) -> X(w) } ;  set An = {w: |Xn(w)-X(w)| > 1/n }

   FIELD:

      i) \Omega in |A
     ii) A\in|A => Ac \in |A
    iii) A,B\in|A => AuB \in |A

   o'-FIELD (=\sigma-ALGEBRA)

    iii) Ai\in|A => Ui Ai \in |A


   Field not o'-Field:  finite & co-finite sets

   o'-Field generated by C:

   Borel Sets

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   countable != infinite