For a continuous r.v. $X$ with CDF $F_{X}$, the **probability density function** (PDF) $f_{X}$ 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 $X$ is located in some small interval around $u$.

## Integrating the PDF to get the CDF

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

$F(u)=∫_{−∞}f(t)dt$This naturally follows from the Fundamental Theorem of Calculus. Notice the usage of two different variables, $u$ and $t$. This is because $t$ is used by the PDF, but $u$ 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 $f(u)≥0$ for all $u∈R$. 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 $(−2,1)$, and therefore integrates to 1 within this range. We can ignore everything else (because we would be integrating zero).