There are at least three PDFs which one may encounter. These are the

Uniform distribution $$\begin{equation*} p(x)=\frac{1}{b-a}\Theta(x-a)\Theta(b-x), \end{equation*}$$ yielding probabilities different from zero in the interval $[a,b]$.

The exponential distribution $$\begin{equation*} p(x)=\alpha \exp{(-\alpha x)}, \end{equation*}$$ yielding probabilities different from zero in the interval $[0,\infty)$ and with mean value $$\begin{equation*} \mu = \int_0^{\infty}xp(x)dx=\int_0^{\infty}x\alpha \exp{(-\alpha x)}dx=\frac{1}{\alpha}, \end{equation*}$$