How does a probability density function differ from a probability mass function?

In probability theory, the primary difference between a probability mass function (PMF) and a probability density function (PDF) lies in the nature of the random variable they describe.

• Probability Mass Function (PMF): Used for discrete random variables, which assume a finite or countably infinite number of values. The PMF, denoted as $f(x)$, provides the actual probability for each specific value of the random variable. The sum of these probabilities across all possible values must equal 1.12
• Probability Density Function (PDF): Used for continuous random variables, which can take on any value within an interval. A PDF does not provide the probability of a specific value; instead, the probability that a variable falls within a given interval is found by calculating the integral of the function over that interval. The total area under the curve of a PDF must equal 1.32

While both functions are non-negative and serve to describe probability distributions, the PMF uses summation to determine probabilities, whereas the PDF relies on integration.12