Definition 5.3.17. Percentiles for Random Variables.
For \(0 \lt p \lt 1\text{,}\) the \(100p^{th}\) percentile is the largest random variable value c that satisfies
\begin{equation*}
F(c) = p.
\end{equation*}
For continuous random variables over an interval \(R = [a,b]\text{,}\) you will solve for c in the equation
\begin{equation*}
\int_a^c f(x) dx.
\end{equation*}
For discrete random variables, it is unlikely that a particular percentile will land exactly on one of the elements of \(R\) but you will want to take the smallest value in \(R\) so that \(F(c) \ge p\text{.}\)
The 50th percentile (as before) is also known as the median.