Spring 2005
John Rust
Economics 425
University of Maryland
Primer on Probability, Random Variables, and Stochastic Processes
1
Basic Probability Theory
These notes provide a basic introduction to
probability theory.
Having a rudimentary understanding of
probability theory, particularly the topic of
random variables,
is essential to be able to understand the
economic topics of 1) choice under uncertainty, and 2) dynamic programming, that are covered in Econ
425.
Definition 1:
A
probability space
(
P
,
Ω
,
F
)
is a mapping
P
:
F
from a
sigma algebra
F
, a collection of
subsets of the
sample space
Ω
, to the
[
0
,
1
]
interval. The sample space
Ω
can be viewed as the
universe
of all possible events, or “states of nature”
and the sigma algebra
F
can be viewed as a
collection of
events
(subsets of possible states of nature). The sigma algebra
F
has to satisfy the following properties:
S1.
Ω
∈
F
S2.
/0
∈
F
(i.e. the empty set,
/0
is in
F
)
S3. if
A
is a member of
F
, then its complement,
A
c
, must also be a member of
F
(i.e. we can write
this statement mathematically as
A
∈
F
=
⇒
A
c
∈
F
).
S4. if
A
∈
F
and
B
∈
F
, then
A
∪
B
∈
F
and
A
∩
B
F
(i.e. if
A
and
B
are events in
F
, the union and
intersection of these sets are also events in
F
),
S5. if
{
A
i
}
is a possibly infinite collection of events with each
A
i
∈
F
, then
∪
∞
i
=
1
A
i
∈
F
and
∩
∞
i
=
1
A
i
∈
F
(i.e. the union and complement of an infinite sequence of events in
F
is also in
F
).
The probability function
P
:
F
→
[
0
,
1
]
must satisfy
P1.
P
(
Ω
) =
1
P2.
P
(
/0
) =
0
P3. if
A
and
B
are in
F
and
A
∩
B
=
/0
, then
P
(
A
∪
B
) =
P
(
A
)+
P
(
B
)
P4. if
A
and
B
are in
F
and
A
∩
B
=
/0
, then
P
(
A
∪
B
)
≤
P
(
A
)+
P
(
B
)
P5. if
{
A
i
}
is an infinite sequence of events in
F
, then
P
(
∪
∞
i
=
1
)
≤
∑
∞
i
=
1
P
(
A
i
)
Thus,
P
(
A
)
can be enterpreted as the probability of the event
A
occuring, and
P
(
A
∪
B
)
is the prob
ability of the event “
A or B
” and
P
(
A
∩
B
)
is the probability of the event “
A and B
” and
P
(
A
c
)
is the
probability of the event “not
A
” (i.e. the event that event
A
did
not
occur). We let the singleton element
ω
∈
Ω
denote the
actual state of the world.
Since the world is very complex (i.e. the state of the world
might involve describing the positions, masses, charges/energies, and momentum of every particle in
the universe) we typically do not directly observe the actual state of the world
ω
. However we can tell
whether certain events occurs, which are sets of states of nature that all result in the same (observable)
1
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outcome. Thus, if
ω
∈
A
(i.e. event
A
contains the true state of nature
ω
), then this is what we mean
by saying, “event
A
has occurred”. An example event
A
might be “it is raining in College Park at noon
Saturday, March 12, 2005”. There are many different states of the world (as described by the exact spec
ification of all atoms and molecules in the universe) that are consistent with event
A
occuring. However
as long any one
ω
occurs so that it rains in College Park at noon on March 12, 2005, then event
A
has
occurred.
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 Spring '08
 Hulten
 Probability, The Lottery, Probability theory, CDF

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