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14.30
Introduction
to
Statistical
Methods
in
Economics
Lecture
Notes
14
Konrad
Menzel
April
2,
2009
Conditional
Expectations
Example 1
Each
year,
a
firm’s R&D
department produces
X
innovations according
to
some
random
process,
where
E
[
X
] = 2
and
Var(
X
) = 2
.
Each
invention
is a
commercial
success with
probability
p
= 0
.
2
(assume independence).
The
number of commercial
successes in
a
given
year are
denoted
by
S
.
Since
we
know
that the
mean of
S
∼
B
(
x, p
) =
xp
, conditional
on
X
=
x
innovations in
a
given
year,
xp
of
them
should
be
successful on average.
The
conditional
expectation
of
X
given
Y
is
the expectation of
X
taken over
the conditional
p.d.f.:
Definition 1
E
[
Y X
] =
�
∞
yf
Y

X
(
y

X
)
if
Y
is discrete
y
yf
Y

X
(
y X
)
dy
if
Y
is
continuous

−∞

Note
that
since
f
Y

X
(
y

X
) carries
the
random variable
X
as
its
argument,
the conditional
expectation is
also a random
variable. However, we can
also define the conditional
expectation of
Y
given a
particular
value
of
X
,
yf
Y

X
(
y x
)
if
Y
is discrete
y
E
[
Y X
=
x
] =
�
∞
yf
Y

X
(
y


x
)
dy
if
Y
is
continuous

−∞
which is just
a number for
any given
value of
x
as
long
as
the conditional
density is
defined.
Since the calculation goes exactly like before, only that we now integrate over the
conditional
distribution,
won’t
do a numerical example (for
the problem set,
just apply definition).
Instead let’s
discuss
more
qualitative
examples
to
illustrate the difference between conditional
and unconditional
examples:
Example 2
(The
Market
for
”Lemons”)
The
following
is a
simplified
version
of a
famous model
for
the
market for
used
cars by the
economist George
Akerlof. Suppose
that there
are
three
types
X
of used
cars:
cars
in an excellent state
(”melons”),
averagequality cars (”average” not in
a
strict, statistical,
sense),
and
cars in a poor condition (”lemons”). Each
type
of car is equally frequent, i.e.
1
P
(”
lemon
”) =
P
(”
average
”) =
P
(”
melon
”) =
3
The seller and
a buyer have the following (dollar) valuations
Y
S
and
Y
B
, respectively, for each
type
of cars:
Type
Seller
Buyer
”Lemon”
5,000
$
6,000
$
”Average”
6,000
$
10,000
$
”Melon”
10,000
$
11,000
$
1