Inverse Distribution Functions

For a distribution function F mapping $\left(\vphantom{ -\infty ,\infty }\right.$ - ∞,∞$\left.\vphantom{ -\infty ,\infty }\right)$ into $\left[\vphantom{ 0,1}\right.$0, 1$\left.\vphantom{ 0,1}\right]$, the inverse distribution function G performs the corresponding inverse mapping from (a subset of) $\left[\vphantom{ 0,1}\right.$0, 1$\left.\vphantom{ 0,1}\right]$ into $\left(\vphantom{ -\infty ,\infty }\right.$ - ∞,∞$\left.\vphantom{ -\infty ,\infty }\right)$; that is, G$\left(\vphantom{ F\left( x\right) }\right.$F$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$$\left.\vphantom{ F\left( x\right) }\right)$ = x and F$\left(\vphantom{ G\left( \alpha \right) }\right.$G$\left(\vphantom{ \alpha }\right.$α$\left.\vphantom{ \alpha }\right)$$\left.\vphantom{ G\left( \alpha \right) }\right)$ = α. Equivalently,

Prob$\displaystyle \left[\vphantom{ X\leq G\left( \alpha \right) }\right.$XG$\displaystyle \left(\vphantom{ \alpha }\right.$α$\displaystyle \left.\vphantom{ \alpha }\right)$$\displaystyle \left.\vphantom{ X\leq G\left( \alpha \right) }\right]$ = F(x) = α

Note that the value that is exceeded with probability α is given by the function G(1 - α). This function is also of interest.

Prob$\displaystyle \left[\vphantom{ X\leq G\left( 1-\alpha \right) }\right.$XG$\displaystyle \left(\vphantom{ 1-\alpha }\right.$1 - α$\displaystyle \left.\vphantom{ 1-\alpha }\right)$$\displaystyle \left.\vphantom{ X\leq G\left( 1-\alpha \right) }\right]$ = F(x)
         = 1 - α = 1 - Prob$\displaystyle \left[\vphantom{ X\leq G\left( \alpha \right) }\right.$XG$\displaystyle \left(\vphantom{ \alpha }\right.$α$\displaystyle \left.\vphantom{ \alpha }\right)$$\displaystyle \left.\vphantom{ X\leq G\left( \alpha \right) }\right]$

In Scientific Notebook, when cumulative distribution functions are named $\limfunc$FunctionDist, then the inverse cumulative distribution functions are named $\limfunc$FunctionInv. For example, $\limfunc$NormalInv is the name of the inverse cumulative distribution function for the normal distribution.