Why are probability distributions denoted with a tilde? What is the meaning of the tilde when specifying probability distributions? For example:
$$Z \sim \mbox{Normal}(0,1).$$
 A: 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)

A: 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.
