In the continuous case, the PDF does not directly depict the actual probability. Instead we define the probability for the stochastic variable to assume any value on an infinitesimal interval around \( x \) to be \( p(x)dx \). The continuous function \( p(x) \) then gives us the density of the probability rather than the probability itself. The probability for a stochastic variable to assume any value on a non-infinitesimal interval \( [a,\,b] \) is then just the integral $$ \begin{equation*} \mathrm{Prob}(a\leq X\leq b) = \int_a^b p(x)dx. \end{equation*} $$ Qualitatively speaking, a stochastic variable represents the values of numbers chosen as if by chance from some specified PDF so that the selection of a large set of these numbers reproduces this PDF.