What does this notation mean? ( $X=\inf\{ x : U \le F(x) \}$ )

Question

What does ( $X= \inf\{ x : U \le F(x) \}$ ) mean?

I know the concept of inf, found here

I found the notation in the title at the end of the chapter on the Inverse Transform Sampling Method where $F(x)$ is the CDF cumulative density function of X and $U$ a random number generate with Uniform distribution $[0,1]$

Answer 1

$\newcommand{\R}{\mathbb R}$ Not sure how much you know about probability, but random numbers like $U$ are formally defined as a function from a probability space $(\Omega,\mathcal F,P)$ to $(\R,\mathcal B(\R)$. You can see more about this here.

Here the probability space is $(\R,\mathcal B(\R))$ and $X$ is a random variable from $\R\to\R$, formally $X(\omega)=\inf\{x:U(\omega)\le F(x)\}$ for $\omega\in\R$.

Answer 2

$U$ is a number and $F$ is a function of one variable. When you put a number $x$ into that function, you get a number $F(x).$ Whether $U\le{}$that number depends of course on what $F(x)$ is, and that depends on what $x$ is. Thus you may put in one number $x$ and find that $U\le F(x)$ and another value of $x$ and find that $U\not\le F(x).$ This distinguishes some values of $x$ from others. Thereby it defines a set of certain values of $x.$ The notation in the question refers to the infimum, i.e. the greatest lower bound, of that set.

For example, suppose $F(x) = 1 - e^{-x}$ for $x\ge0$ and $U= 0.75.$ Then \begin{align} & \{x : U \le F(x)\} = \{x : x\ge \ln 4\} = [\ln4,+\infty) \end{align} and that set has an infimum.