The conditional probability
that a random variable
takes values between
and
conditioned on the event that a random variable
takes values
between
and
, is defined as
 |
(4.4) |
provided the denominator is different from zero. The conditional
probability of events of the kind
reflects our
opinion of which values are more plausible than others, given that
another random variable
has taken a certain value. If
is
independent of
, the probabilities of
are not influenced by
a priori knowledge about
. It holds:
As
goes to 0 in equation (3.4), the
left side of equation (3.4) converges
heuristically to
. In the case of a
continuous random variable
having a density
, the left
side of equation (3.4) is not defined since
for all
. But, it is possible to give a sound
mathematical definition of the conditional distribution of
given
. If the random variables
and
have a joint
distribution
, then the conditional distribution has the
density

for
and
otherwise. Consequently, it holds
The expectation with respect to the conditional distribution can
be computed by
The function
is called the conditional expectation of
given
. Intuitively, it is the expectation of the random
variable
knowing that
has taken the value
Considering
as a function of the random variable
the conditional expectation of
given
is obtained:
is a random variable, which can be regarded as a
function having the same expectation as
. The conditional
expectation has some useful properties, which we summarize in the
following theorem.
Theorem 4.1
Let

be real valued continuous random variables having a
joint density.
- a)
- If
are independent, then
- b)
- If
is a function of
, then
In general, it holds for random variables of the kind
- c)
- The conditional expectation is linear, i.e. for any
real numbers
it holds:
- d)
- The law of iterated expectations:
.
The concept of the conditional expectation can be generalized
analogously for multivariate random vectors
and
Let
be a sequence of chronologically ordered random
variables, for instance as a model of daily stock prices, let
and
, then the
conditional expectation
represents the expected stock price of the following day
given the stock prices
of the
previous
days. Since the information available at time
(relevant for the future evolution of the stock price) can consist
of more than only a few past stock prices, we make frequent use of
the notation
for the expectation of
given the information available up to time
. For all
,
denotes a family of events (having the structure
of a
-algebra, i.e. certain combinations of events of
are again elements of
)
representing the information available up to time
consists of events of which it is known whether they occur up
to time
or not. Since more information unveils as time
evolves, we must have
for
, see Definition 5.1. Leaving out the exact
definition of
we confine to emphasize
that the computation rules given in Theorem 3.1,
appropriately reformulated, can be applied to the general
conditional expectation.