Math 307/607 Supplementary Theory Notes
These notes lay out the axioms of probability and build up several key properties from these axioms.
The
purpose of these notes is to show how the properties can be proved from the axioms.
To understand the
notes it is important to connect them with the illustrations and discussions in class—i.e, these notes are
not “stand alone.”
We have said in class that a formal probability model has 3 ingredients—the sample space
,
Ω
which is a
set of all elementary outcomes to the chance experiment we are modelling, a collection
Σ
of subsets of
,
Ω
which is the collection of all compound outcomes we can handle, and a probability measure
,
P
which is a function assigning to each event its probability; thus
[ ]
:0
,
1
.
P
Σ→
In class and in the book we go through many examples of such models.
Here we lay out the axioms
governing probability spaces, and prove a few simple properties which must follow.
We take as known
set theory concepts such as intersection, union, disjoint etc—these are discussed in class and in the text.
Axiom 1.
If
12
,
AA
are events, then so is the intersection
.
∩
More generally, if
{}
1
n
n
A
∞
=
is a
sequence (or countable collection) of events, then the intersection
1
n
n
A
∞
=
∩
is also an event.
Axiom 2.
If
A
is an event, then so is the complement
.
c
A
Axiom 3.
Ω
is an event.
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 Fall '08
 Luikonnen
 Set Theory, Probability, Probability theory

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