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 thatfor any interval .
In the definition above the quantity 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:
then, we use the conditional density formula:
Let's make an example.
Suppose that the joint probability density function of and is
The support of (i.e., the set of its possible realizations) is
When , the marginal pdf of is
When , the marginal pdf of is because and its integral is zero.
By putting the two pieces together, we obtain
Thus, the conditional pdf of given is
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:
More details about the conditional probability density function can be found in the lecture entitled Conditional probability distributions.
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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.
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