inverseCDF

The inverse cumulative distribution function, often denoted as inverseCDF, quantile function, or percent-point function (PPF), is a function that maps probabilities to values of a random variable. For a given probability $p$ (where $0 \le p \le 1$), the inverseCDF returns the smallest value $x$ such that the cumulative distribution function (CDF) of the random variable is greater than or equal to $p$. In mathematical terms, if $F(x)$ is the CDF of a random variable $X$, then the inverseCDF, denoted as $F^{-1}(p)$, is defined as $F^{-1}(p) = \inf \{x \in \mathbb{R} \mid F(x) \ge p\}$.

The inverseCDF is closely related to the CDF. While the CDF tells us the probability that a random variable will take on a value less than or equal to a certain point, the inverseCDF answers the question: "What value corresponds to a given probability?" This makes it useful in various applications, such as generating random numbers from a specific distribution, determining percentiles, and performing statistical inference.

For continuous probability distributions, the inverseCDF is the inverse of the CDF function. For discrete distributions, where the CDF is a step function, the inverseCDF is defined as described above using the infimum. One of the primary uses of the inverseCDF is in the inverse transform sampling method, a technique for simulating random variates from any probability distribution given its inverseCDF.