Economics 251a
Fall 2006
Basic Probability
1 States of the World
De
f
nition
Let
S
be a set of “states of the world.” Let
γ
s
>
0
be the probability of each
s
∈
S
,
so
X
s
∈
S
γ
s
=1
.
The pair
(
S, γ
)
is called a probability space.
Given any subset
E
⊂
S
,wede
f
ne
γ
(
E
)
≡
X
S
∈
E
γ
s
.
¥
For example, let
S
be the set of all 36 ordered pairs
(
i, j
)
where
i
and
j
are integers between
1and6
.Let
γ
s
/
36
for each
s
∈
S
. We might think of
S
as the possible displays from a roll
of two dice.
A picture of the dice state space is given below:
123456
1
2
3
4
5
6
Each state
s
corresponds to an entry in the above matrix. Let
E
be the states in which the
f
rst die shows 3. Then
E
=
{
(
i, j
):
i
=3
}
=
{
(3
,
1)
,
(3
,
2)
,
(3
,
3)
,
(3
,
4)
,
(3
,
5)
,
(3
,
6)
}
.T
h
u
s
γ
(
E
)=6
/
36 = 1
/
6
.
A random variable
X
on
S
assigns a number
X
s
to each state
s
. We shall often think of
X
s
as
the payo
f
from asset
X
in state
s
. For example,
X
s
could be the face of the
f
rst die,
X
(
i, j
)
≡
i
for all
(
i, j
)
∈
S
. Similarly,
Y
s
could be the face of the second die,
Y
(
i, j
)=
j
. Notice that a state
s
determines the outcome of both random variables. The random variable
X
is constant along
each row, while the random variable
Y
is constant down each column. We can de
f
ne a third
random variable
Z
on
S
by
Z
s
≡
X
s
+
Y
s
for all
s
∈
S
.Thu
s
Z
is the total rolled by both dice.
Notice that
Z
is constant along all southwest—northwest diagonals of the state space. If
u
:
R
→
R
is any function on the reals, then
u
(
X
)
is a random variable if
x
is, de
f
ned by
u
(
X
)
s
=
u
(
X
s
)
.
Again we emphasize that each state
s
determines the outcomes for all random variables (such as
X
,
Y
,
Z, u
(
X
)
)de
f
ned on
S
.
1