Cumulative distribution function and quantiles

The cumulative distribution function of a continuous random variable \(X\) has been defined as

\[ F(x) \;\;=\;\; P(X \le x) \]

and is a monotonic increasing function rising from 0 to 1. Its inverse is

\[ F^{-1}(y) \;\;=\;\; q_y \qquad \text{where }P(X \le q_y) = y \]

Therefore \(F^{-1}(y)\) returns the \(y\)'th quantile of the distribution. Note also that

\[ F\left(F^{-1}(y)\right) \;\;=\;\; y \]

Applying the CDF as a transformation

Transforming a variable into a rectangular distribution

If a continuous random variable \(X\) has cumulative distribution functions \(F(x)\), then the random variable \(Y = F(X)\) has a \(\RectDistn(0, 1)\) distribution.

(Proved in full version)

The converse of this theorem is also useful.

Transforming a rectangular variable into an arbitrary distribution

If \(F(x)\) is a monotonic continuous function of \(x\) rising from 0 to 1, with inverse function \(F^{-1}(\cdot)\), and \(Y \sim \RectDistn(0, 1)\), then the random variable \(X = F^{-1}(Y)\) has a distribution with cumulative distribution function \(F(x)\).

(Proved in full version)

We now illustrate these results with an example.

Example: Exponential distribution

If \(X \sim \ExponDistn(\lambda)\), then it has cumulative distribution function

\[ F(x) \;\;=\;\; 1 - e^{\large -\lambda x}\]

The first result above means that

\[ Y \;\;=\;\; 1 - e^{\large -\lambda X} \;\;\sim\;\; \RectDistn(0,\;1) \]

The inverse function to \(F(x)\) is

\[ F^{-1}(y) \;\;=\;\; -\frac {\log(1 - y)}{\lambda}\]

Therefore if \(Y \sim \RectDistn(0,\; 1)\), then \(X = -\dfrac {\log(1 - Y)}{\lambda}\) has a distribution with cumulative distribution function \(F(x)\) that is therefore an \(\ExponDistn(\lambda)\) distribution.