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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.

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statBeginner
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    I have never hear the term density used in the context of CDF. I would stick with distribution to avoid ambiguity. – copper.hat May 16 '15 at 22:40
  • Also see: http://en.wikipedia.org/wiki/Cumulative_density_function – mhp May 16 '15 at 23:16
  • Right. i saw the wikipedia link before. I wanted to know what the contradiction is. May be with the help of an example. – statBeginner May 16 '15 at 23:47
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    It is more about the use of the language than a counterexample. Density refers to a point and distribution refers to a range. I don't think there is anything more profound to it than that. – mhp May 17 '15 at 01:38
  • The term density in general refers to the amount of something per unit length/area/volume, e.g. mass density, charge density, etc. The probability density function tells you, literally, the density of probability. It is something you integrate over a region to obtain the actual probability. The cumulative distribution function, on the other hand, tells you the actual integrated probability between $-\infty$ and the point in question, so it is not a density: it is not something you can integrate over a region to obtain a sensible quantity. –  May 17 '15 at 01:54

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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 $x$.
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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.

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