# Why are probability distributions denoted with a tilde?

What is the meaning of the tilde when specifying probability distributions? For example:

$$Z \sim \mbox{Normal}(0,1).$$

• Have a look at point 4 of this entry from Wolfram MathWorld.
– user10525
Oct 27, 2012 at 15:40
• @Procrastinator: you should go ahead and submit this as an answer. I don't think it will get any better. Oct 27, 2012 at 19:23
• In statistics it is used over variable to show the median~X Aug 29, 2022 at 13:46
• It's an oddity (to me) that there is nothing like a standard notation for median, but the fact that a tilde over a symbol may be sometimes be used for median, or indeed for other summaries of a variable, has nothing whatsoever to do with this notation. Aug 29, 2022 at 15:34
• Tilde ∼ means "is sampled from this distribution" Jan 9, 2023 at 15:49

The ~ (tilde) used in that way means "is distributed as". Why? To ask why doesn't make much sense to me, its just a convention. To cite Brian Ripley:

Mathematical conventions are just that, conventions. They differ by field of mathematics. Don't ask us why matrix rows are numbered down but graphs are numbered up the y axis, nor why x comes before y but row before column. But the matrix layout has always seemed illogical to me. -- Brian D. Ripley (answering a question why print(x) and image(x) are layouted differently) R-help (August 2004)

• I'll wait and see if someone comes up with an idea about the history or the "why" and if not i'll accept this one
– jsj
Oct 28, 2012 at 2:00

I can't comment on the history, but I believe it might be the following. The ~ symbol is commonly used in mathematics to denote an equivalence relation. In the context of probability theory it is used to denote equivalance in (marginal) distribution. So when we say,

Z ~ N(0,1),

what we mean is that the random variable Z has the same marginal distribution as the random variable N(0,1). (The latter being a standard normal random variable, by definition.) This interpretation requires that you interpret the right-hand-side of the equation as referring to a random variable, not a distribution function. Under this interpretation, the ~ sign means "has the same distribution as". Since this is reflexive, symmetric and transitive, it is an equivalence relation.

• Equivalence relation on what set? There is no such thing as a "set of all random variables."
– whuber
Jan 1, 2013 at 4:07
• One probably could have "something like" equivalence relations in the contect of categories, that is, on proper classes. Apr 5, 2017 at 15:18