# Does there exist a probability distribution could not be associated with a cumulative distribution function?

This wiki page says

A continuous probability distribution is a probability distribution with a cumulative distribution function that is absolutely continuous. Equivalently, it is a probability distribution on the real numbers that is absolutely continuous with respect to Lebesgue measure. Such distributions can be represented by their probability density functions. If the distribution of X is continuous, then X is called a continuous random variable. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.

It seems that the first sentence in the quotation implies that there exists some other types of probability distribution that could not be associated with a cumulative distribution function?

Is my understanding right, or I am over reading the part?

• All random variables $X$ have a cumulative distribution function (CDF) defined by $$F_X(x) = P[X\leq x] \quad \forall x \in \mathbb{R}$$ The CDF function is not always continuous. I would say a random variable $X$ is "continuous type" if its CDF function is continuous. I would not complexify by requiring an "absolutely continuous" CDF. The advantage of adding the "absolutely" is that it allows you to also have a PDF. – Michael Oct 13 '19 at 4:36

Every probability distribution on $$\mathbb R$$ is associated to a cumulative distribution function (and every non-decreasing function $$F$$ with $$\inf_x F(x)=0$$ and $$\sup_x F(x)=1$$ is associated to a distribution!). The quoted text is probably using "with" to refer to the whole phrase "a cumulative distribution function that is absolutely continuous."
For the fact that every probability distribution does define a cumulative distribution function, just note that defining, for a probability measure $$\mu$$ on $$\mathbb R$$, the CDF as $$F(x)=\mu((-\infty,x))$$ gives a perfectly good CDF - although there's a little bit of ambiguity about what you should do at point masses where $$(-\infty,x)$$ and $$(-\infty,x]$$ have different measure - but this depends on what you want from a CDF and is a very manageable caveat.
Conversely, given a non-decreasing function on $$\mathbb R$$, you can define a measure with the property that $$\mu((a,b))=\left(\lim_{x\rightarrow b^-}F(x)\right) - \left(\lim_{x\rightarrow a^+}F(x)\right).$$ I won't give all the technical details of this construction, but this measure is called the Lebesgue-Stieltjes measure associated to $$F$$.
There's a slight issue in this association having to do with point-masses, but you can fix that by imposing conditions like right-continuity on $$F$$ which basically decide whether $$F(x)$$ includes a point mass at $$x$$ or not.
• Can I convince you to use $F$ instead of $f$? If so, I will +1 vote! (Just to keep convention of usign $F_X(x)$ to denote CDF and $f_X(x)$ to denote PDF (when it exists)). – Michael Oct 13 '19 at 4:17