Joint probability density function
The concept of joint probability density function (joint pdf) is a multivariate generalization of the concept of probability density function. The joint pdf characterizes the distribution of a continuous random vector. The probability that the realization of a continuous random vector will belong to a given set is equal to the multiple integral of the joint pdf over that set.
Table of contents
Definition
The following is a formal definition.
Definition
Let
be a
continuous random vector. The joint probability density function of
is a function
![[eq1]](/images/joint_probability_density_function__4.png)
such
that![[eq2]](/images/joint_probability_density_function__5.png)
for
any
hyper-rectangle![[eq3]](/images/joint_probability_density_function__6.png)
Example
Let
be a
random vector having joint probability mass
function:![[eq4]](/images/joint_probability_density_function__9.png)
In other words, the joint pdf is equal to
if both components of the vector belong to the interval
![$\left[ 0,1\right] $](/images/joint_probability_density_function__11.png)
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:![[eq5]](/images/joint_probability_density_function__14.png)
More details
Joint probability density functions are discussed in more detail in the lecture entitled Random vectors.
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