Can someone please explain what is inverse CDF function

Question

can someone please explain in simple terms with an example that what is inverse of a CDF function and how do we get it from any CDF value.

Answer 1

Let $F$ be a CDF.

Then define its "inverse" as the function $\Phi:(0,1)\to\infty$ prescribed by:$$u\mapsto\inf\{x\in\mathbb R\mid F(x)\geq u\}$$

A characteristic property is: $$\Phi(u)\leq x\iff u\leq F(x)$$

Consequently if $U$ has uniform distribution on interval $(0,1)$ we have:$$P\left(\Phi(U)\leq x\right)=P\left(U\leq F(x)\right)=F(x)$$

Example:

If $F(x)=1-e^{-x}$ for $x>0$ (so standard exponential distribution) then $\Phi(u)=-\ln(1-u)$.