Final answer:
A probability density function (PDF) cannot be a probability mass function (PMF) as they are used for different types of random variables: continuous and discrete, respectively.
Step-by-step explanation:
The question posed is whether a joint probability density function, used in the context of continuous variables, could be a valid probability mass function, typically associated with discrete variables. To clarify, in probability theory, a probability density function (PDF) is a function that describes the likelihood of a continuous random variable taking on a certain value. In contrast, a probability mass function (PMF) is used for discrete random variables and specifies the probability that a discrete random variable is exactly equal to some value.
As such, a probability density function cannot be a probability mass function because they cater to fundamentally different types of random variables. The role of a PDF is to describe the probability that a continuous random variable falls within a certain range, which is represented by the area under the curve of the function over that range. This is fundamentally different from a PMF that assigns probabilities to exact outcomes of a discrete random variable. Nevertheless, both types of functions must adhere to the rules of probabilities, meaning the total probability (or integral in the continuous case) must equal one.