For a continuous r.v. with CDF , the probability density function (PDF) is the derivative of the CDF.

It is “analogous” to the PMF for discrete random variables, but it is not a probability. It approximates the probability that is located in some small interval around .

Integrating the PDF to get the CDF

Let be a continuous random variable with PDF . Then, the CDF of is given by

This naturally follows from the Fundamental Theorem of Calculus. Notice the usage of two different variables, and . This is because is used by the PDF, but is used by the CDF. We need two different variables because they represent different things.

Properties of valid PDFs

The PDF is nonnegative. This means for all . This makes sense because probabilities can’t be negative, and we’re integrating over the area.

The PDF integrates to 1 from to . However, usually the PDF is non-zero only in a specific range like , and therefore integrates to 1 within this range. We can ignore everything else (because we would be integrating zero).