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 Rp)
- 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 28=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(Ai)=1/2.
- So:
- General case: n tosses.
;
- Define
- It is possible to prove that
Jargon to come later:
random variables Xi 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 Ck to be the event first head occurs
after k consecutive tails:
where
;
Ac means complement of A.
Our assumption guarantees
Complicated Events: examples
- Strong Law of Large Numbers: for our model
P(A2) = 1.
- In fact,
.
- If
P(A2) = 1 then
P(A1) = 1.
- In fact P(A3)=1 so
P(A2)=P(A1) = 1.
Some mathematical questions to answer:
- 1.
- Do (4) and (5)
determine P(A) for any
? [NO]
- 2.
- Do (4) and (5)
determine P(Ai) 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(Ac) = 1-P(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 X1
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
2000-09-12