The joint probability density function (joint pdf) is a function used to characterize the probability distribution of a continuous random vector.
It is a multivariate generalization of the probability density function (pdf), which characterizes the distribution of a continuous random variable.
The generalization works as follows:
the integral of the density of a continuous variable over an interval is equal to the probability that the variable will belong to that interval;
the multiple integral of the joint density of a continuous random vector over a given set is equal to the probability that the random vector will belong to that set.
The following is a formal definition.
Definition
Let
be a
continuous random vector. The joint probability density function of
is a function
such
that
for
any
hyper-rectangle
The
notationused
in the definition above has the following meaning:
the first entry of the vector
belongs to the interval
;
the second entry of the vector
belongs to the interval
;
and so on.
Furthermore, the
notationwhere
is a
-dimensional
vector is used interchangeably with the
notation
where
are the
entries of
.
Finally, the notation
means that the multiple integral is computed along all the
co-ordinates.
Here are some examples.
Let
be a
random vector having joint
pdf
In other words, the joint pdf is equal to
if both components of the vector belong to the interval
and it is equal to
otherwise.
Suppose we need to compute the probability that both components will be less
than or equal to
.
This probability can be computed as a double
integral:
Let
be a
random vector having joint probability density
function
Suppose we want to calculate the probability that
is greater than or equal to
and at the same time
is less than or equal to
.
This can be accomplished as
follows:where
in step
we have performed an integration by parts.
One of the entries of a continuous random vector, when considered in isolation, can be described by its probability density function, which is called marginal density.
The joint density can be used to derive the marginal density. How to do this is explained in the glossary entry about the marginal density function.
Joint probability density functions are discussed in more detail in the lecture entitled Random vectors.
Previous entry: Joint distribution function
Next entry: Joint probability mass function
Most of the learning materials found on this website are now available in a traditional textbook format.