Probability density function

Definition 11.6 (Probability density function). A non-negative real-valued function $f(x)$ is a probability density function (pdf) of a continuous random variable $X$ if, for every $a\le b$ in the range of $X$,

— i.e. probability over an interval is the area under the curve $y=f(x)$ between $x=a$ and $x=b$.

Theorem 11.2 (characterisation, without proof). $f(\cdot)$ is a pdf for some continuous random variable $X$ if and only if (i) $f(x)\ge0$ for every $x$, and (ii) $\displaystyle\int_{-\infty}^{\infty}f(x)\,dx=1$ (total area under the curve is exactly $1$, the discrete "$\sum f(x_k)=1$" condition translated into area form).