# 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 $$3^{'})$$ ?

• All such "distributions" are mixtures of (a) an ordinary distribution function, (b) a point mass at $-\infty,$ and (c) a point mass at $+\infty.$ This is an immediate consequence of $(3^\prime).$
– whuber
Commented Oct 11, 2021 at 15:44
• @whuber By placing point masses at $+\infty,-\infty$ we avoiding taking the limit? Commented Oct 11, 2021 at 16:31
• Actually, no. We first have to extend the real line by adjoining points $\pm\infty$ at either end. (There is a standard construction in topology--a form of "completion" at the "ends" of the space--to do this.) The masses you assign to these points do not help you evaluate limits concerning what happens on the usual real numbers. Using the extended reals like this can be a helpful way to record information about (say) things that have less than a 100% chance of occurring.
– whuber
Commented Oct 11, 2021 at 17:06