Probability density function

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

$$f_X(u)=F_X^{\prime }(u)$$

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

$$F(u)={\int }_{-\infty }^{u}f(t)dt$$

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 $f(u)\ge 0$ for all $u\in \mathbb{R}$. This makes sense because probabilities can’t be negative, and we’re integrating over the area.

The PDF integrates to 1 from $-\infty$ to $\infty$. 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).