# What does $x \sim X$ mean in probability?

I just want to confirm what $$x \sim X$$ mean in probability. What does the small $$x$$ and big $$X$$ represent? And can we replace $$\sim$$ with $$=$$.

Disclosure: I had read many probability textbooks from front to back such as Leon-Garcia and Papoulis, but never encountered this notation until I looked at some notes online. I think this is a notation used in stastistics, because $$\sim$$ is not very informative in my view.

• It would be helpful to give an example of this symbol being used in context. – Trevor Gunn Oct 9 '18 at 2:10
• I have seen the $\sim$ sign often in probability, but rarely if ever with a lower-case letter on the left-hand side. If you have really seen something like this I would very much want to see an example with some context around it. – David K Oct 9 '18 at 2:29
• @DavidK hi, there is like entire fields in AI and machine learning dedicated to using arbitrary symbols like this arxiv.org/pdf/1406.2661.pdf – Cauchy's Carrot Oct 9 '18 at 2:34
• @DavidK See also arxiv.org/pdf/1711.00141.pdf – Cauchy's Carrot Oct 9 '18 at 2:34
• Ah, well, one thing about CS papers is they often bastardize mathematical notations, and one thing about papers in arxiv is there are no independent reviewers to send them back for revision repeatedly until the authors define their undefined symbols. But the use of $\sim$ at least seems to be intended the way it is explained in the answer below. – David K Oct 9 '18 at 2:45

Typically you would say something like the following

$$X \sim \textrm{Uniform}(-1,1) \tag{1}$$

then $$X$$ is a random variable and it follows the uniform distribution with parameters $$-1$$ and $$1$$. Typically a uniform distribution is like this.

f_{X}(x) =\begin{align}\begin{cases} \frac{1}{b-a} & \textrm{ for } a \leq x \leq b \\ 0 & \textrm{ everywhere else } \end{cases} \end{align} \tag{2}

when with parameters $$-1$$ and $$1$$ we have

f_{X}(x) =\begin{align}\begin{cases} \frac{1}{2} & \textrm{ for } -1 \leq x \leq 1 \\ 0 & \textrm{ everywhere else } \end{cases} \end{align} \tag{3}

In other words, it is saying that $$x$$ is a random variable and following a distribution given by $$X$$ in your case, with a distribution function given by it. In other words, it is like the $$=$$ sign but typically it can refer to a family of functions.

## Edit:

If you read the paper in section $$3$$ on page $$2$$ it literally defines things.

To learn the generator’s distribution $$p_{g}$$ over data x, we define a prior on input noise variables $$p_{z}(z)$$,

It uses notation like this in $$4$$

The generator G implicitly defines a probability distribution $$p_{g}$$ as the distribution of the samples G(z) obtained when $$z ∼ p_{z}$$

## Edits:

For the second part if you read the paper, it is defined on page $$2$$.

where $$p$$ is the distribution to learn and $$q_{\theta}$$ is the distribution defined by the implicit generator. The expectation is minimized over a parametrized class of functions

and you note that $$\otimes$$ is the tensor product.

• But in their papers, $x$ is defined as data, and $p_g$ is distribution. Whereas in your answer, $X$ is a random variable, and $f_X(x)$ is the probability density function. Are you telling me that they mean the same thing? If so, why not written as the way you do (or in most probability textbooks like the ones I have read)? And what is a tensor product of distributions? – Cauchy's Carrot Oct 9 '18 at 3:07
• The third paragraph said, "where p is the distribution to learn and qθ is the distribution defined by the implicit generator." but it didn't say anything about $p \otimes q_\theta$. – Cauchy's Carrot Oct 9 '18 at 3:09
• as the other person noted, they bastardize the notation. en.wikipedia.org/wiki/Tensor_product . I am assuming this may actually be like a covariance matrix? but I don't know. – user3417 Oct 9 '18 at 3:11
• I don't know enough about the area my main point was to read the paper and search through the bibliography. – user3417 Oct 9 '18 at 3:12
• I don't know either, it was just some paper on Reddit that looks interesting but I couldn't get through due to the notation. – Cauchy's Carrot Oct 9 '18 at 3:13