The probability distribution of a continuous random variable can be characterized by its probability density function (pdf). When the probability distribution of the random variable is updated, by taking into account some information that gives rise to a conditional probability distribution, then such a distribution can be characterized by a conditional probability density function.
The following is a formal definition.
Definition
Let
and
be two continuous random variables. The conditional probability density
function of
given
is a function
such
that![[eq2]](/images/conditional-probability-density-function__6.png)
for
any interval
.
In the definition above the quantity
![[eq4]](/images/conditional-probability-density-function__8.png){kind=small}
is
the
conditional
probability that
will belong to the interval
,
given that
.
In order to derive the conditional pdf of a continuous random variable given the realization of another one, we need to know their joint probability density function (see this glossary entry to understand how joint pdfs work).
Suppose that we are told that two continuous random variables
and
have joint probability density function
.
Then, we are also told that the realization of
has been observed and
,
where
denotes the observed realization.
How do we compute the conditional probability density function of
so as to take the new information into account?
This is done in two steps:
first, we compute the
marginal
density of
by integrating the joint
density:![[eq6]](/images/conditional-probability-density-function__20.png){kind=small}
then, we use the conditional density
formula:![[eq7]](/images/conditional-probability-density-function__21.png)
Let's make an example.
Suppose that the joint probability density function of
and
is![[eq8]](/images/conditional-probability-density-function__24.png)
The support of
(i.e., the set of its possible realizations)
is
When
,
the marginal pdf of
is![[eq11]](/images/conditional-probability-density-function__29.png)
When
,
the marginal pdf of
is
because
and
its integral is zero.
By putting the two pieces together, we
obtain![[eq15]](/images/conditional-probability-density-function__34.png)
Thus, the conditional pdf of
given
is![[eq16]](/images/conditional-probability-density-function__37.png)
Note that we do not need to worry about division by zero (i.e., the case when
)
because the realization of
always belongs to the support of
and, as a consequence,
.
We have just explained how to derive a conditional pdf from a joint pdf, but
things can be done also the other way around: if we are given the marginal pdf
and the conditional
,
then the joint distribution can be derived by performing a simple
multiplication:![[eq21]](/images/conditional-probability-density-function__44.png){kind=small}
More details about the conditional probability density function can be found in the lecture entitled Conditional probability distributions.
Previous entry: Binomial coefficient
Next entry: Conditional probability mass function
Please cite as:
Taboga, Marco (2021). "Conditional probability density function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/conditional-probability-density-function.
Most of the learning materials found on this website are now available in a traditional textbook format.
