# Does "probability distribution" refer to the PDF or CDF

In my work and studies, I keep coming across statements that are similar to the following:

Quote from one source:

A better angle, at least from the perspective of GANs, is to define similarity in the sense of probability distribution. Two data sets are considered similar if they are samples from the same (or approximately same) probability distribution. Thus more specifically we have our training data set X ⊂Rn consisting of samples from a probability distribution μ (with density p(x)), and we would like to find a probability distribution ν (with density q(x)) such that ν is a good approximation of μ. By taking samples from the distribution ν we obtain generated objects that are “similar” to the objects in X.

Quote from a different source:

The graph of a continuous probability distribution is a curve. Probability is represented by area under the curve. We have already met this concept when we developed relative frequencies with histograms in Chapter 2. The relative area for a range of values was the probability of drawing at random an observation in that group. Again with the Poisson distribution in Chapter 4, the graph in Example 4.14 used boxes to represent the probability of specific values of the random variable. In this case, we were being a bit casual because the random variables of a Poisson distribution are discrete, whole numbers, and a box has width. Notice that the horizontal axis, the random variable x, purposefully did not mark the points along the axis. The probability of a specific value of a continuous random variable will be zero because the area under a point is zero. Probability is area. The curve is called the probability density function (abbreviated as pdf). We use the symbol f(x) to represent the curve. f(x) is the function that corresponds to the graph; we use the density function f(x) to draw the graph of the probability distribution.

I have been trying to really understand at more than a surface level the mathematics behind generative models, but a roadblock has been trying to determine what authors mean by "probability distribution". I am fairly certain that "probability distribution" in the first quote is referring to the CDF, since the author specifically denotes the density as p(x), which I think refers to the PDF. In the second quote, the author portrays "the curve" is a probability distribution, and taking an integral over it results in a probability, obviously conveying it is the PDF.

Maybe "probability distribution" in fact has no concrete definition and the reader is left to figure out what it is referring to themselves, but it would make my life a lot easier if I knew it referred to one or the other.

• People tend to be a little loose with the term “probability distribution” (and I think this causes genuine confusion). The distribution of a random variable $X$ is the probability measure $\mu$ on $\mathbb R$ defined by $\mu(E) = P(X \in E)$. Here $E$ is a (measurable) subset of $\mathbb R$. Sometimes people use the term “probability distribution” when they really mean CDF or probability density function or probability mass function. Sometimes people use the term probability distribution to mean “probability measure”. It’s unfortunate that such an important term is not used more carefully. Commented Jul 27, 2021 at 16:20
• In my opinion there has been a genuine pedagogical failure regarding the term "probability distribution". Commented Jul 27, 2021 at 21:56
• See this answer of mine over on stats.SE for another take on this question. Commented May 28, 2022 at 22:20