In modern papers in statistics, it is common to use the symbol $E[X]$ to refer to the expectation of a random variable $X$.

While reading (a translated version of) "Convergence Rate of Nonparametric Estimates of Maximum-Likelihood Type" by A. S. Nemirovskii, B. T. Polyak, A. B. Tsybakov, I noticed that they seem to use the notation $M(X)$ to refer to the expectation integral.

I tried Google translating "expectation", "mean", "average" and "integral" to Russian, but none of these translations seem to begin with $M$.

I'd be interested in finding out what the $M$ stands for and when this convention was first introduced.

  • 4
    $\begingroup$ I don't think its recent. ru.wikipedia.org/wiki/… seems to indicate its M for the Russian form of "mathematical expectation" or "mittlewert" or "mean". $\endgroup$ – kimchi lover Nov 19 '19 at 18:00
  • $\begingroup$ Thanks, i really should have checked the russian wikipedia. $\endgroup$ – cdwe Nov 19 '19 at 18:07

One would think that Russian usage stems from Kolmogorov's seminal works on probability. However, in Über die Summen durch den Zufall bestimmter unabhängiger Größen (1928) he uses $\mathfrak{M}$ to denote probability (presumably from messen, measure), not mathematical expectation. In the famous Grundbegriffe der Wahrscheinlichkeitsrechnung (1933), which gave the modern axiomatization of probability theory, mathematical expectation is called die mathematische Erwartungen and denoted $\mathsf{E}$. Unfortunately, its 1936 Russian translation does not seem to be available online, and the 1974 edition, which is and uses $\mathsf{M}$, can not be trusted since it says in the preface that notation has been modernized. Khinchin (Kolmogorov's student) also uses $\mathsf{E}$ in Математические основания статистической механики (Mathematical foundations of statistical mechanics, written in 1941).

But even the English translation of Gnedenko-Kolmogorov's Limit distributions for sums of independent random variables (1954, Russian original 1949) uses $\mathsf{M}$ and features математическое ожидание (mathematical expectation) prominently around the definition. Mittlewert or mean (среднее) do not start with м in Russian, are not mentioned nearby, and Kolmogorov's previous usage does not make motivation by a foreign spelling likely. Late 1940-s in Russia were characterized by promotion of Russian nationalism and condemnation of "rootless cosmopolitans", making it even less likely. Gnedenko's Курс теории вероятностей (Course of Probability Theory, 1950), which became the standard text on the subject in Russian universities, introduces $\mathsf{M}$ in a similar way. So even in the unlikely event that Gnedenko and Kolmogorov had something else in mind, $\mathsf{M}$ for Mатематическое (ожидание) is what Russian readers would have taken away from the text.

Speaking generally, motivation of the original user is not necessarily the motivation for the subsequent widespread adoption of a notation. In Kolmogorov's case, $\mathsf{E}$ comes from German spelling. That it happens to match English "Expected" is just a coincidence, but it may well have played a decisive role in its adoption in the Anglophone literature. His $\mathfrak{M}$ for probability was not similarly adopted.

The Sources of Kolmogorov’s Grundbegriffe by Shafer and Vovk is a good source on Kolmogorov's early work on probability.

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