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

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

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    $\begingroup$ Have a look at point 4 of this entry from Wolfram MathWorld. $\endgroup$ – user10525 Oct 27 '12 at 15:40
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    $\begingroup$ @Procrastinator: you should go ahead and submit this as an answer. I don't think it will get any better. $\endgroup$ – Stephan Kolassa Oct 27 '12 at 19:23

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)

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    $\begingroup$ 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 $\endgroup$ – jsj Oct 28 '12 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.

  • $\begingroup$ Equivalence relation on what set? There is no such thing as a "set of all random variables." $\endgroup$ – whuber Jan 1 '13 at 4:07
  • $\begingroup$ One probably could have "something like" equivalence relations in the contect of categories, that is, on proper classes. $\endgroup$ – kjetil b halvorsen Apr 5 '17 at 15:18

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