Probability Density Function (PDF)

Definition

A probability density function $f_X(x)$ of a continuous random variable $X$ is a function satisfying:

  1. $f_X(x) \geq 0$ for all $x$.
  2. $\displaystyle\int_{-\infty}^{\infty} f_X(x)\, dx = 1$.

Probabilities are obtained by integration:

$$ \mathbb{P}(a \leq X \leq b) = \int_a^b f_X(x)\, dx. $$

Note that $f_X(x)$ is not a probability — it can exceed 1. Only integrals of $f_X$ over intervals yield probabilities.