Quantile function

Definition

Let $F \colon \mathbb{R} \to [0, 1]$ be the cumulative distribution function of a real random variable $X$. The quantile function of $X$ is the function $Q \colon (0, 1) \to \mathbb{R}$ with $$Q(p) = \inf \{x \in \mathbb{R} : F(x) \ge p\}.$$ Here $\inf$ is the infimum. The number $Q(p)$ is the $p$-quantile of $X$.

Remarks

The set in the definition is nonempty and bounded below for $0 < p < 1$, and the infimum is a minimum because $F$ is right-continuous. So $F(Q(p)) \ge p$, and $Q(p) \le x$ if and only if $p \le F(x)$. If $F$ is continuous and strictly increasing, then $Q = F^{-1}$. If $U$ is uniform on $(0, 1)$, then $Q(U)$ has the cumulative distribution function $F$. This is inverse transform sampling. $Q(1/2)$ is a median of $X$, and $Q(1/4)$ and $Q(3/4)$ are the lower and upper quartiles.

Explanations

L1 human

The quantile function answers the question "below which value does a fraction $p$ of the probability lie?". For the exponential distribution with rate $1$, $F(x) = 1 - e^{-x}$, so $Q(p) = -\ln(1 - p)$.