Lecture 2 on BST 631: Statistical Theory I – Kui Zhang, 08/21/2008
1
Review for the previous lecture
Definition:
sample space, event, operations (union, intersection, complementary), disjoint, pairwise disjoint
Theorem:
commutativity, associativity, distribution law, DeMorgan’s law
Probability theory:
sample space, sigma algebra, probability function
Theorem:
define a probability function on finite and countable infinite sample spaces
Chapter 1 – Probability Theory
Chapter 1.2 – Basic of Probability Theory
1.2.1 Axiomatic Foundations
Theorem 1.2.6:
Let
1
{ ,
,
}
n
S
s
s
=
"
be a finite set. Let
B
be any sigma algebra of subsets of
S
. Let
1
,
,
n
p
p
"
be
nonnegative numbers that sum to 1. For anyA
∈
B
, define
(
)
P A by
:
(
)
i
i
i s
A
P A
p
∈
=
∑
. Then P is a probability
function on
B
. This remains true if
1
2
{ ,
,
}
S
s s
=
"
is a countable set.
Calculating the probability of an Event:
The following steps can be used to find a probability of an event from a
countable sample space:
1.
Define the experiment, determine the possible outcomes, and construct the sample space
S
.
2.
Define probability function
P
on
S
.

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