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Transcript
Probablity Models have 3 components:
Sample Space (Ω)
Events on Sample Space (F)
Functions defined on the sample space (P)
⇒Probability Model: (Ω,F,P)
Why consider sets? Probability models involve
families of sets - combining and operating on a
family of sets, using unions, intersections,
complements. The results of such operations
must then also be in the family
Set Algebra Results: Cont’d
Recall: Sample Space
Subsets, Union, Intersection
Cummutative , Associative, Distributive Laws
Recall: A partition of Ω is any collection of disjoint sets
A α that fill out the whole space: ie U A α = Ω,
and A α A β = ∅, α ≠ β
{ }
Recall: An event is a subset of Ω. All Events are
subsets - but all subsets are not events. (Here, we
want to assign probabilities to events in a coherent
fashion - cannot always do this if all subsets are
events.
A partition of Ω serves to partition an event in Ω, ie if
{A α }is a partition of Ω (from before) and let B be
an event, the sets BA α constitute a partition of B:
(
)(
)
U BA α = B, and BA α BA β = ∅, α ≠ β
Related Result on ascending/descending sequences:
{
}
The sequence A n + 1 − A n constitutes a partition of
their union, A= U A n
{ }
Ex: Consider the disks Bn defined by
{ }
x 2 + y 2 ≤ 1 − 1 / n, n = 1, 2,...The sequence Bn is
(?).Bn − Bn −1 are rings between concentric circles
which constitute a partion of their union (the
closed disk)
Definition: A Field is a family of sets closed under
complementation, finite unions, and finite
intersections.
Functions on a sample space: X(ω ) maps from Ω to a
new set. The “image point” is the value of X(ω)
corresponding to ω. For a given value of X (ie x),
the set of ω ‘s such that X(ω)=x is the pre-image of
x. A set of values of X - say B, defines a pre-image
set in Ω . Pre-image sets may or may not be
events.
A function defined on a sample space induces a
partition of the sample space.
Recall: Probability Axioms:
Given a sample space Ω, a collection of subsets called
events, the Probability of an event E is P(E). The
set function P must satisfy the Axioms of
Probability:
1.
2.
3.
For every event E, 0 ≤ P(E) ≤ 1
P(Ω) = 1
If EF= E∩F= 0, then P(E∪F) = P(E) +P(F)
Related Probability Results on ascending/descending
sequences:
{ }
Suppose En an ascending sequence: E1, E 2 ,....En ,
Ei ⊂ E j,, i<j, the P(U En ) = lim P(En )
n→∞
{ }
{ }
Likewise, En is a descending sequence, En ,
E1, E 2 ,....En , Ei ⊂ E j,,the i>j, then
P(I En ) = lim P(En )
n→∞
{ }
Axiom 3a: For any sequence Ei such that Ei E j =0 for
n  ∞
i ≠ j, P  U E i  = ∑ P E i
 i=1  i=1
( )
This is called the axiom of countable additivity.
Law of Total Probability
If {En} is a countable partition of Ω, then
n
P(F) = ∑ P(FEn )
i=1
Outcomes of Symmetric Experiments:
Given a total of N experiments, if outcomes are
equally likely (random), then the odds
"probability"of one outcome is 1/N. (lottery or coin
tossing examples). The Probability of an event E,
P(E) = 1/N
Law of Large Numbers:
Suppose repeat a long sequence of trials and the
results of any set do not depend (independent) on
results of other trials, the proportion of
occurrences of an event stabilizes and tends to a
limiting value.
Consider probability to be the long run limit of
proportions of "successes."
Ex: Consider a baseball player’s succie times “at bat”
as successive trials of an experiment. Batting
average (no hits/no times at bat) fluctuates, but
tends to a limiting value.
Ex: Tosses of a coin - consider the proportion of
“heads”