�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
1
14.30
Introduction
to
Statistical
Methods
in
Economics
Lecture
Notes
5
Konrad
Menzel
February
19,
2009
We
distinguish
2
different types
of
random
variables:
discrete and continuous.
Discrete
Random
Variables
Definition 1
A random variable
X
has a
discrete
distribution if
X
can take on only a
finite (or countably
infinite) number
of
values
(
x
1
,x
2
,...
)
.
Definition 2
If random
variable
X
has a
discrete
distribution, the
probability density function
(p.d.f.)
of
X
is defined
as the
function
f
X
(
x
) =
P
(
X
=
x
)
If
{
x
1
,x
2
,...
}
is
the set of
all possible values
of
X
,
then for
any
x /
∈ {
x
1
,x
2
,...
}
,
f
X
(
x
) = 0.
Also
∞
f
X
(
x
i
) = 1
i
=1
The probability that
X
∈
A
for
A
⊂
R
is
P
(
X
∈
A
) =
f
X
(
x
i
)
x
i
∈
A
Example 1
If
X
is the
number we
rolled
with
a
die, all
integers
1
,
2
,...,
6
are
equally likely.
More
generally,
we
can
define
the
discrete
uniform
distribution
over the
numbers
x
1
,x
2
,...,x
k
by its p.d.f.
1
f
X
(
x
) =
k
if
x
∈ {
x
1
,x
2
,...,x
k
}
0
otherwise
This corresponds to the
simple
probabilities for an
experiment with
sample
space
S
=
{
x
1
,x
2
,...,x
k
}
.
Example 2
Suppose
we
toss 5 fair coins independently from another and
define
a
random variable
X
which
is
equal
to
the
observed number of
heads.
Then
by our counting
rules,
n
(
S
) = 2
5
= 32
, and
5
n
(”k heads”) =
,
using
the
rule
on combinations. Therefore
k
5
1
1
5
1
5
P
(
X
= 0) =
=
,
P
(
X
= 1) =
=
0
32
32
1
32
32
,
5
1
10
5
1
10
P
(
X
= 2) =
=
,
P
(
X
= 3) =
=
2
32
32
3
32
32
,
5
1
5
5
1
1
P
(
X
= 4) =
=
,
and
P
(
X
= 5) =
=
.
4
32
32
5
32
32
1