[Note: although this question has an accepted answer, the investigation is not finished yet. I encourage you to post your findings.]

Who first introduced the notation "$X \sim Q$", meaning that $Q$ is the probability distribution for $X$, and its related meanings?

Classical texts such as Jeffreys's Theory of Probability (3rd ed. 1961, 2nd ed. 1948) and Fisher's Statistical Methods for Research Workers (13th repr. ed. 1963) and Statistical Methods and Scientific Inference (1956) do not seem to use it (in Jeffreys "$\sim$" denotes the logical not).

I checked also the references given in this informative answer, but did not find anything relevant there.

(This notation is by no means universal in recent times or today, see eg Mosteller & Tukey's Data Analysis and Regression (1977) or Jaynes's Probability Theory (2003). In fact it does not make much sense when probability is seen as an extension of formal logic (eg Keynes, Johnson, Jeffreys, Pólya, Hailperin, Jaynes), defined over propositions rather than random variables.)

Edit (2021-06-24): The earliest reference I found so far is a paper by Khatri written in 1965 and published 1967, where the notation is introduced on p. 1854. The earliest textbook reference found so far is by Srivastava & Khatri, 1979, p. 41.

John Aldrich also kindly replied to my inquiry, saying that he never investigated the origin of this notation.

  • $\begingroup$ Thank you @MattF. , good point. I added some dates and texts $\endgroup$
    – pglpm
    Jun 24 at 16:44
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    $\begingroup$ If it is of any help I did not see the tilde notation in Dixon & Macy’s “Introduction to Data Analysis” 2nd edition (1957) but it was in Y.A. Rozanov’s “Probability: A Concise Course” revised English edition (1969) $\endgroup$
    – daszlosek
    Jun 27 at 6:04
  • $\begingroup$ Thank you @daszlosek . Are you sure about Rozanov's? I checked it, and "$\sim$" seems to be used as "approximately equals", not as "has distribution". For example on page 38 top, §7; and p. 55, §10, eqn (5.5). $\endgroup$
    – pglpm
    Jun 27 at 9:27
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    $\begingroup$ I checked H.A. David’s Order Statistics, in the first edition of 1970, and the tilde notation is not there. $\endgroup$
    – Matt F.
    Jun 28 at 22:47
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    $\begingroup$ A few comments up, the allusion is to Dixon and Massey, in its time a fine introductory text, but with a different title, and not one I would expect to be formal with its notation. Interesting question: wish I had a further answer. $\endgroup$
    – Nick Cox
    Sep 16 at 7:59

Early uses

There are some earlier uses since 1961 by Ingram Olkin with others.

  • Olkin, Ingram, and Robert F. Tate. "Multivariate correlation models with mixed discrete and continuous variables." The Annals of Mathematical Statistics (1961): 448-465.

    $X \sim F(x)$ means that $x$ is distributed according to the d.f. $F(x)$, and $x(n) \to F(x)$ means that the asymptotic d.f. of $x(n)$ is $F(x)$.

  • Olkin, Ingram, and Herman Rubin. "A characterization of the Wishart distribution." The Annals of Mathematical Statistics (1962): 1272-1280.

    We write $X \sim \mathcal{W}(\Lambda,p,n), (n > p-1, \Lambda: p \times p, \Lambda > 0 )$, to mean that $X$ is a $p \times p$ symmetric matrix with the density function $$p(X) = c \vert \Lambda \vert ^{n/2} \vert X \vert^{(n-p-1)/2} e^{-(\frac{1}{2}) \text{tr} \Lambda X} $$

digging further you come across the notation in the dissertation "Multivariate Tests of Hypotheses with Incomplete Data" by Raghunandan Prasad Bhargava (student of Olkin) in 1963

We use the notation $Y \sim p(z)$ to mean that the probability density of $Z$ is $p(z)$, and $Y \underset{H}{\sim} p(z)$ to denote that the probability density of $Z$ under the hypothesis $H$ is $p(z)$.

Srivastava is also an alumnus of Stanford university (and used the notation in a paper of 1965).

Before 1961

  • Before this time Olkin uses the more verbose descriptions of the form 'Let ... be independent and each distributed as ...'. So it seems like he either started it around 1961 or picked it up from others.

  • For Rubin there is neither earlier use of $\sim$ in this way. In one paper he seems to use it to indicate asymptotic equality. (analogous to Bachmann–Landau notations)

    Rubin, Herman. "The estimation of discontinuities in multivariate densities, and related problems in stochastic processes." Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics. University of California Press, 1961.

  • Other uses of the symbol $\sim$ are by Kolmogorov in

    Колмогоров, Андрей Николаевич. "Об аналитических методах в теории вероятностей." Успехи математических наук 5 (1938): 5-41.

    I do not know Russian but based on google translate I guess that equation 67 $$P^p_k \sim \frac{A^kp^k}{k!}e^{-Ap}$$ is supposed to mean is approximately distributed as

  • A similar use is in

    Колмогоров, Андрей Николаевич. "Обобщение формулы Пуассона на случай выборки из конечной совокупности." Успехи математических наук 6.3 (43 (1951): 133-134.

    In which we have an equation $$P_n(m \vert N,M) \sim P_n^\star(m\vert N , M)$$ and $P^\star$ is a simpler expression that is supposed to approximate to $P$ (in this case the hypergeometric distribution is being approximated)

  • Another use by Kolmogorov is in

    Колмогоров, Андрей Николаевич. "Об операциях над множествами." Математический сборник 35.3-4 (1928): 415-422.

    In which the following equation occurs $$\overline{\overline{X}} \sim X$$ Again, I do not know Russian and have to decipher it. To me it seems that it means that the operation denoted by $\overline{\overline{X}}$ will give the same result as the operation denoted by $X$. (this is not anymore about probability).

  • Paul Lévy uses the tilde symbol also as 'approximation' or 'asymptotic formula'. For instance in

    Lévy, Paul. "Sur le développement en fraction continue d'un nombre choisi au hasard." Compositio mathematica 3 (1936): 286-303.

    La probabilité de l’existence de $p$ valeurs supérieures à $n$ dans la suite $y_1, y_2, ..., y_n$ est d’autre part $$C_{n}^p \left( \frac{1}{n}\right)^p\left( 1-\frac{1}{n}\right)^{n-p} \sim \frac{1}{ep!} \quad (n \to \infty)$$

    But, here both sides of the binary relation are mathematical expressions. It is not a 'statistical variable' on the left side.

  • Lévy uses $\sim$ in stochastic expressions in

    Lévy, Paul. "Wiener's random function, and other Laplacian random functions." Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, 1951.

    Now it is an asymptotic relation for the case of infinitesimally small steps. For instance Lévy's fundamental stochastic infinitesimal equation (equation 2.1.1) is written as $$\delta X(t) \sim dt \int_{t_0}^t F(t,u) dX(u) + \zeta \sigma(t) \sqrt{dt}$$ and the symbol is explained as

    The symbol $\sim$ means that, as $dt \to 0$, the two first moments of the difference of the two sides are $o(dt)$ or $o\left[d\omega(t) \right]$ [if $\sigma^2(t)dt$ is replaced by $d\omega(t)$].

(in Jeffreys "$∼$" denotes the logical not)

The tilde notation for negation dates at least back to use by Giuseppe Peano in 1987. See Jeff Miller's webpage: https://jeff560.tripod.com/set.html

  • $\begingroup$ "In one paper he seems to use it to indicate asymptotic equality." It's still extensively used to denote this in analytic number theory. en.wikipedia.org/wiki/Prime_number_theorem for example. $\endgroup$ Sep 15 at 23:43
  • $\begingroup$ @MatthewDrury it fits in the big-$O$ and small-$o$ notations. Small-$o$ means that $f_n/g_n$ approaches zero, big-$O$ that it is bounded and the tilde means that it approaches one. en.wikipedia.org/wiki/… $\endgroup$ Sep 15 at 23:49
  • $\begingroup$ Thank you for these very early references! I wonder whether Olkin borrowed that notation from some further sources. In the ISO standard on mathematical notation, the symbol "~" means "asymptotically equal to" – and this seems to be its standard use in the probability & statistics papers before 1961 (except Jeffrey's and his logic use). You could write to John Aldrich about your findings. $\endgroup$
    – pglpm
    Sep 16 at 7:33
  • $\begingroup$ @pglpm I did look into the circle around Olkin to see if this use might have been borrowed from others and you do see that there is some use of it at Stanford university but all of that I found is from after 1961. It is also interesting to note that the tilde symbol is explicitly explained in the article's of the early 60's and that indicates that it was at least a new thing at that time. I might add these and more considerations that point to Olkin in a separate paragraph. A definite proof might be found by digging into his letters and papers. I saw that there is a big library available. $\endgroup$ Sep 16 at 10:21
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    $\begingroup$ You're right. Besides Olkin, it can be Tate as well. Interesting is this article from 1966 where he is explaining the notation very accurately. Besides the $\sim$ symbol he also used the $\approx$ symbol to denote asymptotic similarity (that sounds like he has been thinking a lot about it but only the $\sim$ got stuck). Too bad it is difficult to find more about the works from the University of Washington and Oregon where he has been. I guess that I had been too much focussed on Stanford University for which it is easier to track down changes in notation $\endgroup$ Sep 17 at 13:48

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