# What is the correct notation for stating that random variables X and Y are independent?

Often people put down $X\perp\ Y$ as independence, but this merely means that the expectation of X and Y is zero and does not have any implication on correspondence between their joint PDF, CDF and marginal PDF, CDF, etc...

Is there a widely agreed upon notation for saying that two RVs are independent?

• \perp\!\!\!\perp yields $\perp\!\!\!\perp$ which is often used to denote independence, e.g. $X\perp\!\!\!\perp Y$. – Stefan Hansen Oct 6 '14 at 9:54

$$\require{txfonts}$$As you say, the use of $$\perp$$ (\perp) for independence is not good, since it often means orthogonal, which in probabilitry theory translates to correlation zero. Independence is a (much) stronger concept, so needs a stronger symbol, and sometimes I have seen $$\perp\!\!\!\perp$$ (\perp\!\!\!\perp) used. That seems like a good idea!
OK, seems like math markup here does not like \Perp, but it is defined in $$\LaTeX$$ packages pxfonts/txfonts. It is like \perp, but with double vertical lines. Above I replace a hack.
Apart from multivariate normal distributions of the kind $(X,Y)$, where one can write $Cov(X,Y)=0$, one writes "$X$ and $Y$ are independent". Why bother with symbols if normal language is already clear and short?
• It is only for jointly normal random variables that $\operatorname{Cov}(X,Y)=0$ can be used as a substitute for saying $X$ and $Y$ are independent. Is there much saving in stating "$X$ and $Y$ are jointly normal random variables with $\operatorname{Cov}(X,Y)=0$ and means $\mu_X, \mu_Y$ and variances $\sigma_X^2, \sigma_Y^2$ respectively" versus "$X\sim N(\mu_X,\sigma_X^2)$ and $Y\sim N(\mu_Y,\sigma_Y^2)$ are independent random variables"? – Dilip Sarwate Oct 6 '14 at 13:59
$X|Y = X$ does not reflect the symmetry of the (non-)relation but shouldn't it signify independence?
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• I read this as a definite answer--but it's dangerous to pose answers as questions, for people misinterpret the intention and vote to close them. Regardless, there is an interesting distinction operating here: "$X|Y=X$" may indeed imply independence, but it appears to do so by virtue of a mathematical argument: it does not in itself signify independence (unless you actually define independence in terms of conditional distributions, which is a bit unusual). – whuber Jun 2 '15 at 17:30