This lecture discusses how to factorize the
joint probability
density function of two continuous random variables (or random vectors)
and
into two factors:
the
conditional
probability density function of
given
;
Table of contents
The factorization, which has already been discussed in the lecture entitled Conditional probability distributions, is formally stated in the following proposition.
Proposition (factorization)
Let
be a continuous random vector with support
and joint probability density function
.
Denote by
the conditional probability density function of
given
and by
the marginal probability density function of
.
Then,![[eq5]](/images/factorization-of-joint-probability-density-functions__14.png){kind=small}
for
any
and
.
When we know the joint probability density function
and we need to factorize it into the conditional probability density function
and the marginal probability density function
,
we usually proceed in two steps:
marginalize
by integrating it with respect to
and obtain the marginal probability density function
;
divide
by
and obtain the conditional probability density function
(of course this step makes sense only when
).
In some cases, the first step (marginalization) can be difficult to perform.
In these cases, it is possible to avoid the marginalization step, by making a
guess about the factorization of
and verifying whether the guess is correct with the help of the following
proposition.
Proposition (factorization
method)
Suppose there are two functions
and
such that
for any
and
,
the following
holds:![[eq17]](/images/factorization-of-joint-probability-density-functions__32.png){kind=small}
for any fixed
,
,
considered as a function of
,
is a probability density function
Then,
The proof covers the case in which
and
are random variables. The proof for the case in which they are random vectors
is a straightforward generalization of this proof. The marginal probability
density of
satisfies![[eq20]](/images/factorization-of-joint-probability-density-functions__40.png){kind=small}
therefore,
by property 1
above:![[eq21]](/images/factorization-of-joint-probability-density-functions__41.png)
where
the last equality follows from the fact that, for any fixed
,
,
considered as a function of
,
is a probability density function and the integral of a probability density
function over
equals
.
Therefore,![[eq23]](/images/factorization-of-joint-probability-density-functions__47.png){kind=small}
which,
in turn,
implies![[eq24]](/images/factorization-of-joint-probability-density-functions__48.png)
Thus, whenever we are given a formula for the joint density function
and we want to find the marginal and the conditional functions, we have to
manipulate the formula and express it as the product of:
a function of
and
that is a probability density function in
for all values of
;
a function of
that does not depend on
.
Example
Let the joint density function of
and
be![[eq26]](/images/factorization-of-joint-probability-density-functions__58.png)
The
joint density can be factorized as
follows:where![[eq28]](/images/factorization-of-joint-probability-density-functions__60.png)
and![[eq29]](/images/factorization-of-joint-probability-density-functions__61.png)
Note
that
is a probability density function in
for any fixed
(it is the probability density function of an
exponential random variable with parameter
).
Therefore,
Please cite as:
Taboga, Marco (2021). "Factorization of joint probability density functions", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/factorization-of-joint-probability-density-functions.
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