* Housekeeping Details

- Lec: Mon/Wed 2:50-4:05
- OH:  Dunno yet
- HW:  Approx 6 probs/wk; reviewed but not graded in detail
- Txt: 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); 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)
       
       E[X]:    Expectation of SOME (maybe not all) random variables X

   Big Ideas in Probability:

       LLN:

       If { X_i } are Indep Identically-Distributed RV's with same mean
       \mu = E[ X_i ], then

            (1/n)  \sum_{i<=n}  X_i   ->   \mu

       [ what does it MEAN for a sequence RANDOM VARIABLES like 

             Y_n = (1/n) \sum  X_i

       to "converge" to a constant \mu or to a random variable Y??? ]

       CLT:

       If { X_i } are IID with same mean \mu and finite variance 
       \sig^2=E[(X_i-\mu)^2], then

             Z_n = \sqrt {n} * (Y_n - \mu)/sig  ==>  No(0,1)
 
       [ what does it MEAN for a sequence of DISTRIBUTIONS to converge?? ]

       MCT:

       If  X_n  =  E[ X_{n+k} | X_1,...,X_n ] for every k>=0 and n, then
       UNDER SOME CONDITIONS (what?  why?),

           X_n  ->  X

       for some random variable X, and for SOME random times T,

                   E[ X_T | Info up to time n ]  =  X_n

       [ what does it MEAN to find expectation "given" some "info" ? ]

----------------------

   Operations:

        Complement;  A^c  =  "not A"  =  {w: w \notin A}
            
        Union over arbitrary index set;

             \cup A_\alpha  =  {w:  w \in A_\alpha for at least one \alpha }

             A \cup B       =  "A or B (or perhaps both)"

        Intersection over arbitrary index set

             \cap A_\alpha  =  {w:  w \in A_\alpha for all \alpha }

             A \cap B       =  =  A B  =  "A and B both"

        Set difference; symmetric difference;

             A /\ B  =  (A \cap B^c) \cup (A^c \cap B)
               -- 
                     =  A\B u B\A   =  "exactly one of A, B occurs"
   Relations:

        containment:  A \subset B  :  "A implies B"

        disjoint:     A B = \emptyset

        equality;     A = B:          "A if-and-only-if B"

   De Morgan's Law:

        (\cup A_i)^c  =  \cap (A_i ^c)
        (\cap A_i)^c  =  \cup (A_i ^c)

   Indicator Functions: 1_A

   Lim Inf:  union of intersections = EA  =  \cup_n \cap_{k>=n}  A_k
   Lim Sup:  intersection of unions = IO  =  \cap_n \cup_{k>=n}  A_k

   Note  (Lim Inf)  \subset  (Lim Sup); sometimes they coincide,
         but not always.  Some examples, with \Omega = |N:

         A_n = {n,n+1,...}            LimSup = LimInf = emptyset

         A_n = {1,2,...,n}            LimSup = LimInf = |N

         A_{2n} = Evens, A_{2n+1} = Odds:  LimSup = |N, LimInf = {}



   Example:

       a_n -> a  <==>  V eps>0 \exists N<oo s.t. n>N -> |a_n-a| < eps

       {w: Xn(w) -> X(w) } ;  set An = {w: |Xn(w)-X(w)| > 1/n },
                              consider LimInf An and LimSup An.

   Not every subset A of \omega will be an "event", but we will need
   to show that some are.  Here are some rules:

   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

--------------

   countable != infinite  (present Cantor arg if time allows)