# How to check whether a function is a valid cumulative distribution function?

This is very related to this and this but not exactly what I'm looking for. From Wikipedia I know that the following 4 conditions must hold for a function to be a CDF:

Every cumulative distribution function $$F_X$$ is non-decreasing and right-continuous. Furthermore, the following must hold:

$$\lim_{x\to -\infty}F_X(x)=0$$ and $$\lim_{x\to +\infty}F_X(x)=1$$

I know now what must hold but not how to apply/check it. Can someone provide a minimal example of a function that is a CDF with the necessary steps on how to check it? I couldn't find what I'm looking for elsewhere.

• You prove it has the 3 properties above and that $F(x)=\mathbb{P}_X(-\infty,x]$ – badatmath May 10 '19 at 19:20
• @badatmath There are 4 properties, so which 3? Also, my question is on how to prove it? What methods do you use? How do you apply them...etc. – Does it matter May 10 '19 at 19:22
• What do you mean by "how to check" the four conditions? Those conditions are not phrased in the terminology of probability. Those are just conditions on a function. E.g. to check the two limits, you just evaluate the limits and prove they are $0, 1$ respectively. – antkam May 10 '19 at 21:05

If you prove it's a distribution function for $$X$$, meaning proving the following $$F_X(x)=\mathbb{P}_X((-\infty,x])=\mathbb{P}(X\in (-\infty,x])=\int1_{(-\infty,x]}(t) \mathbb{P}_X(dt)=\int_\Omega 1_{(-\infty,x]}(X) \mathbb{P}(d\omega).$$ then you are done, since the three other properties follow from continuity of measures and the fact that a measure is monotone and for a probability measure $$\mathbb{P}(\Omega)=1$$.
If you prove that $$F$$ is non decreasing: $$x\geq y \Rightarrow F(x)\geq F(y)$$, right continuous $$\lim_{y\downarrow x}F(y)=F(x)$$ and the two limits at $$\pm \infty$$, then you showed it is a distribtion function. To prove it is a distribution function for $$X$$, you need to know something about $$X$$. Like it's distribution or density or something..