The symbol $\perp\!\!\!\!\perp$ in statistics is a way to denote statistical independence of a collection of random variables. I have seen two forms of it. The first is highly suitable in writing about independence between two variables:

$$X\perp\!\!\!\!\perp Y$$

The second works for two variables, but is especially suited for multiple variables:

$$\perp\!\!\!\!\perp \left[ X_1, \cdots, X_n \right]$$

What these symbols mean, for a collection of (let's say real-valued) random variables $\{ X_j \}_{j=1}^n$, is

$$\perp\!\!\!\!\perp \left[ X_1, \cdots, X_n \right] \iff F_{X_1, \cdots, X_n} (x_1, \cdots, x_n) = \prod_{j=1}^n F_{x_j}(x_j)$$

where $F_{X_1, \cdots, X_n}$ is the joint cumulative distribution function and the $F_{x_j}$ is the marginal cumulative distribution function for the $j$th random variable.

What is the earliest-known publication that used the $\perp\!\!\!\!\perp$ symbol (either as a binary or n-ary relation) to denote statistical independence between/among random variables?

  • $\begingroup$ I occasionally see $X_1 \perp\!\!\!\!\perp \cdots \perp\!\!\!\!\perp X_n$, or similar, so the former notation is capable of illustrating the n-ary case. $\endgroup$
    – Galen
    May 17, 2022 at 21:29
  • $\begingroup$ Note that I am asking about $\perp\!\!\!\!\perp$, which has two vertical bars, rather than $\perp$ which has one vertical bar. $\endgroup$
    – Galen
    May 17, 2022 at 21:35

1 Answer 1


An early reference that uses this notation and (at a minimum) introduced many people to this notation is the following publication.

Dawid, A. P. “Conditional Independence in Statistical Theory.” Journal of the Royal Statistical Society. Series B (Methodological) 41, no. 1 (1979): 1–31. http://www.jstor.org/stable/2984718.

Have you found earlier uses?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.