MATH 235A – Probability Theory
Lecture Notes, Fall 2009
Part I: Foundations
Dan Romik
Department of Mathematics, UC Davis
Draft version of 10/14/2009
Lecture 1: Introduction
1.1
What is probability theory?
In this course we’ll learn about probability theory. But what exactly
is
probability theory?
Like some other mathematical fields (but unlike some others), it has a dual role:
•
It is a
rigorous mathematical theory
– with definitions, lemmas, theorems, proofs
etc.
•
It is a
mathematical model
that purports to explain or model reallife phenomena.
We will concentrate on the rigorous mathematical aspects, but we will try not to forget the
connections to the intuitive notion of reallife probability. These connections will enhance
our intuition, and they make probability an extremely useful tool in all the sciences. And
they make the study of probability much more
fun
, too!
A note of caution is in order,
though: Mathematical models are only as good as the assumptions they are based on. So
probability can be
used
, and it can be (and quite frequently is)
ab
used...
1
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1.2
The algebra of events
A central notion in probability is that of the
algebra of events
(we’ll clarify later what
the word “algebra” means in this context).
We begin with an informal discussion.
We
imagine that probability is a function, denoted
P
, that takes as its argument an “event”
(i.e., occurrence of something in a reallife situation involving uncertainty) and returns a
number in [0
,
1] representing how likely this event is to occur. For example, if a fair coin is
tossed 10 times and we denote the results of the tosses by
X
1
, X
2
, . . . , X
10
(where each of
X
i
is 0 or 1, signifying “tails” or “heads”), then we can write statements like
P
(
X
i
= 0) = 1
/
2
,
(1
≤
i
≤
10)
,
P
10
X
i
=1
X
i
= 4
!
=
(
10
4
)
2
10
.
Note that if
A
and
B
represent events (meaning, for the purposes of the present informal
discussion, objects that have a welldefined probability), then we expect that the phrases
“
A
did not occur”, “
A
and
B
both occurred” and “at least one of
A
and
B
occurred” also
represent events. We can denote these new events by
¬
A
,
A
∨
B
, and
A
∧
B
, respectively.
Thus, the set of events is not just a set – it is a set with some extra structure, namely
the ability to perform
negation
,
conjunction
and
disjunction
operations on its elements.
Such a set is called an
algebra
in some contexts.
But what if the coin is tossed an
infinite
number of times?
In other words, we now
imagine an infinite sequence
X
1
, X
2
, X
3
, . . .
of (independent) coin toss results. We want to
be able to ask questions such as
P
(infinitely many of the
X
i
’s are 0)
=
?
P
lim
n
→∞
1
n
n
X
k
=0
X
k
=
1
2
!
=
?
P
∞
X
k
=0
2
X
k

1
k
converges
!
=
?
Do such questions make sense?
(And if they do, can you guess what the answers are?)
Maybe it is not enough to have an
informal
discussion to answer this...
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 Fall '11
 DanRomik
 Probability, Probability theory, Ω, Durrett, Borel

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