Examples of Defective Distributions I was reading about Stochastic Convergence and I came across a term called Defective Distribution.
Essentially what they refer to as a {Defective Distribution is a distribution that has all the properties of a Cumulative Distribution $F$, i.e.,
$1)$ $F:\mathbb{R}\rightarrow[0,1]$
$2)$  Based on https://en.wikipedia.org/wiki/Cumulative_distribution_function it is also upwards continuous monotonic increasing function
BUT
the Defective Distribution does not have the property of the Cumulative Distribution
$3)$ $\lim_{x \to -\infty}F(x)=0$ and  $\lim_{x \to \infty}F(x)=1$
the Defective Distribution has
$3^{'})$ $\lim_{x \to -\infty}F(x)\geq0$ and  $\lim_{x \to \infty}F(x)<1$

I would like to ask if there is a know such distribution, i.e. a know Defective Distribution with the property $3^{'})$ ?

Well, certainly.
For instance, the arctangent is monotonically increasing from $\lim_{x\to-\infty}\arctan x=-\frac{\pi}{2}$ to $\lim_{x\to\infty}\arctan x=\frac{\pi}{2}$. So something like
$$ F: x\mapsto \frac{1}{2}+\frac{1-2\epsilon}{\pi}\arctan x $$
will satisfy $\lim_{x\to-\infty}F(x) = \epsilon$ and $\lim_{x\to\infty}F(x) = 1-\epsilon$ for any (small, but) positive $\epsilon$.

R code:
xx <- seq(-10,10,by=.1)
epsilon <- 0.1
plot(xx,1/2+(1-2*epsilon)*atan(xx)/pi,
  type="l",las=1,ylim=c(0,1),xlab="x",ylab="F(x)")
abline(h=c(epsilon,1-epsilon),lty=2)

