Postscript version of this file
STAT 870 Lecture 1
Course outline
Goals of Today's Lecture:
 Motivate probability modelling
 Define
 Probability Space
 Random variables (in R^{p})
 The distribution of a random variable
 Cumulative Distribution Function
Course outline:
 Measure theoretic foundations of probability:
 fields
 Measurability arguments
 Formal definition of expected value
 Fatou's lemma, monotone convergence theorem, dominated
convergence theorem.
 Modes of convergence: in probability, in mean square, almost sure.
Course outline continued:
 Statements of some of the following famous theorems of probability:
 Weak law of large numbers
 Strong law of large numbers
 Lindeberg central limit theorem
 Martingale convergence theorems
 Ergodic theorems
 Renewal theorem
Course outline continued:
 One week introductions to each of:
 Markov Chains
 Poisson Processes
 Point Processes
 Birth and Death Processes
 Brownian motion and diffusions
 Renewal processes
 Student presentations
Models for coin tossing
 Probability modelling: select family
of possible probability measures.
 Make best match between mathematics, real world.
 interpretation of probability: long run limiting relative
frequency
 Coin tossing problem: many possible probability measures on .
 For n=3,
has 8 elements and 2^{8}=256 subsets.
 To specify P: specify 256 numbers. Generally impractical.
 Instead: model by listing some assumptions about P.
 Then deduce what P is, or how to calculate P(A)
 three approaches to modelling coin tossing:
 Counting model:

(1) 
Disadvantage: no insight for other problems.
 Equally likely elementary outcomes: if
and
are two singleton sets in then P(A)=P(B). If
,
say
then
So
and (1) holds.
 Defect of models: infinite
not easily handled.
 Toss coin till first head. Natural
is set of all sequences of k zeros
followed by a one.
 OR:
.
 Can't assume all elements equally likely.
 Different approach: model using independence:
Coin tossing example: n=3.
 define
and
 Then
 Note P(A)=1/8,
P(A_{i})=1/2.
 So:
 General case: n tosses.
;
 Define
 It is possible to prove that
Jargon to come later:
random variables X_{i} defined by
are independent.
 This is basis of most common modelling tactic:
assume

(2) 
and that for any set of events of the form given above

(3) 
 Motivation: long run relative frequency interpretation plus
assumption that the outcome
of one toss of the coin is incapable of influencing the outcome
of another toss.
 Advantages: generalizes to infinite .
 Toss coin infinite number of times:
is an uncountably infinite set. Model assumes for any nand any event of the form
with each
we have

(4) 
 For a fair coin add
the assumption that

(5) 
 Is P(A) determined by these assumptions??
 Consider
where
.
Our assumptions
guarantee
 In words, our model specifies that the first n of our infinite
sequence of tosses behave like the equally likely outcomes model.
 Define C_{k} to be the event first head occurs
after k consecutive tails:
where
;
A^{c} means complement of A.
Our assumption guarantees
Complicated Events: examples
 Strong Law of Large Numbers: for our model
P(A_{2}) = 1.
 In fact,
.
 If
P(A_{2}) = 1 then
P(A_{1}) = 1.
 In fact P(A_{3})=1 so
P(A_{2})=P(A_{1}) = 1.
Some mathematical questions to answer:
 1.
 Do (4) and (5)
determine P(A) for any
? [NO]
 2.
 Do (4) and (5)
determine P(A_{i}) for i=1,2,3?
[YES]
 3.
 Are (4) and (5) logically consistent?
[YES]
Probability Definitions
Probability Space (or Sample Space): ordered
triple
.
Axioms guarantee can compute probabilities by usual rules, including
approximation without contradiction.
Consequences:
 1.
 Finite additivity if
pairwise disjoint:
 2.
 For any event A
P(A^{c}) = 1P(A) .
 3.
 If
are events then
 4.
 If
then
Most subtle point is field,
.
Needed to avoid some contradictions
which arise if you try to define P(A) for every subset A of when
is a set with uncountably many elements.
Random Variables:
Vector valued random variable: function X,
domain ,
range in
such that
is defined for any constants
.
Notation:
and
is shorthand for an event:
X function on
so X_{1}
function on .
Formal definitions:
The Borel field in
is the smallest
field in
containing every open ball
(To see that
there is, in fact, such a ``smallest'' field you prove the
following assertions:
 1.
 The intersection of an arbitrary family of fields is
a field. [Homework set 1.]
 2.
 There is at least one field of subsets of
containing
every open ball. [Homework set 1.]
Now define the Borel field in
to be
where the intersection runs over all fields which contain every open ball.)
Richard Lockhart
20000912