STT 861: Theory of Probability and
Statistics I
0
Introduction
According to Wikipedia,
statistics is a mathematical science pertaining to
the collection, analysis, interpretation or explanation, and presentation of
data.
It has applications to a wide variety of academic disciplines, from
the natural and social sciences to the humanities, government and business.
Since the quantities that we observe are most of the time
random
we need a
theory about the likelihood of potential events and the underlying mechanics
of random systems.
This is what
probability theory
is all about.
In this
sequence of two courses (STT 861 and 862) we will first study the basic
theory of probability and random variables and then use that to learn about
statistical methods.
1
Probability
1.1
Sample Spaces and Events
Probability is a model that assigns to each
event
A
a number,
P
(
A
), de
scribing the likelihood of this event to occur. An event is just a collection of
possible outcomes, and the collection of
all
possible outcomes (the biggest
event) is called
the sample space
, often denoted by Ω.
Example 1.1
(Tossing a coin)
.
The sample space is Ω =
{
H, T
}
and the
event that “the outcome is head” is
{
H
}
.
Example 1.2
(Throwing a die)
.
The sample space is Ω =
{
1
,
2
,
3
,
4
,
5
,
6
}
and the event that “the outcome is even” is
{
2
,
4
,
6
}
.
Example 1.3
(Driving to work)
.
When Lisa drives to work, she passes
through three intersections with traffic lights.
At each light, she either
stops,
s
, or continues,
c
. In this example,
the sample space, Ω =
{
ccc, ccs, css, csc, sss, ssc, scc, scs
}
,
“she does not stop at all”=
{
ccc
}
,
“she has to stop in all the lights’=
{
sss
}
,
“she stops in at most one light”=
{
ccc, ccs, csc, scc
}
and so on.
More examples are available in the textbook. Please read them.
1
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1.2
Probability Measure
Probability measure (or simply probability) is a map that assigns a real
number
P
(
A
) to each event
A
. It must form a consistent model, as described
in the following set of requirements (axioms):
1. For every event
A
,
P
(
A
)
≥
0.
2. For the sample space (biggest event) Ω,
P
(Ω) = 1.
3. For pairwise disjoint (mutually exclusive) events
A
1
, A
2
, . . .
,
P
∞
[
i
=1
A
i
!
=
∞
X
i
=1
P
(
A
i
)
.
As a visual aid,
mentally
interpret probabilities as areas with the use of
Venn diagrams. Please read the textbook for a quick review of set theory
and Venn diagrams. More specifically, read the commutative laws, the asso
ciative laws, and the distributive laws of the set operations. There are two
more laws called
de Morgan’s Laws
:
1. (
A
∪
B
)
c
=
A
c
∩
B
c
, and
2. (
A
∩
B
)
c
=
A
c
∪
B
c
.
Example 1.4
(An easy example)
.
16 universities are taking part in a bas
ketball tournament. In the first round there are 8 games (1 vs. 2, 3 vs. 4,
. . .
, 15 vs. 16). An outcome is a list of winners, e.g. (2
,
3
,
5
,
7
,
9
,
12
,
13
,
16).
An event is a set of several outcomes; the sample space has 2
8
= 256 out
comes. Setting how likely each outcome is, we can compute
P
(
A
) by adding
up the likelihoods of the outcomes comprising the event
A
.
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 Fall '08
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 Probability theory, CDF

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