Percentiles of a Continuous Distribution

Let $p$ be a number between $0$ and $1$. The $(100p)$-th percentile or $p$-th quantile of the distribution of a continuous rv $X$, denoted by ${\eta }_p$, is defined implicitly by the equation:

Assuming we can find the inverse of $F(x)$, this can also be written as:

In particular, the median of a continuous distribution is the 50th percentile ${\eta }_{.5}$ or $F^{-1}(.5)$. We'll sometimes just denote this as $\eta$.