Why is Cumulative "Density" wrong? CDF stands for cumulative distribution function. However, it is "loosely" referred to as Cumulative Density many times. As i write this question, I have a suggestion toolbar on this page that lists over 10 questions with the words "Cumulative Density" in them. 
I came across this question in this forum post where a comment clearly highlights how the word "cumulative" contradicts "denisty" and "cumulative density function" is a term that shouldnt be used. However, I came across the term in many other posts and even answers in this forum like here and 
here.
Unfortunately, I do not have privileges to comment on the post which suggests the difference between the two. Hence a new question. Can someone explain the contradiction in details please. Thank you.
 A: I haven't heard "cumulative density" used before. People are saying that it is a contradiction.  To see why, it helps to look at the definitions of cumulative distribution function (CDF) and probability density function (PDF).
Assume $X$ is a continuous random variable.
The CDF is 
$$F(x) = P(X\leq x) = \int_{-\infty}^x f(x) \, dx.$$
It is the integral of the PDF $f(x)$ up to some value $x$.  


*

*The CDF $F(x)$ is "cumulative" because it accumulates the total area under the PDF from $-\infty$ to $x$. The CDF says something about $X$ over an entire interval $(-\infty,x]$.

*The PDF $f(x)$ is the "density" because it tells us how how likely it is that $X$ will be near $x$.  More quantitatively, $f(x) \, dx \approx P(x < X < x + dx)$. The PDF says something about $X$ locally at each point $x$.

A: Let $(\Omega, \mathcal F, \mathbb P) $ be a probability space and $X:\Omega\to\mathbb R$ a random variable. The distribution function of X is $$F(x) = \mathbb P\circ X^{-1}(-\infty,x] = \mathbb P(X \leqslant x). $$ If $F$ is a continuous function, in which case $F$ is actually absolutely continuous (due to monotonicity and boundedness), so F is differentiable almost everywhere. A continuous function f which extends F' is called a probability density function of X. 
