According to this Wikipedia entry, "Mu was derived from the Egyptian hieroglyphic symbol for water, which had been simplified by the Phoenicians and named after their word for water". So, my question is, why the early statisticians decided to use the letter Mu to denote the population mean / expected value? Was it because that letter was not occupied for denoting other concepts at the time or there was a deeper idea behind the choice?
The letters that derive from $\mu$ include the Roman M and the Cyrillic М. Hence considering that the word "mean" starts with an $m$ the choice seems relatively straightforward given an already existing tradition to use greek letters in mathematical abbrevation.
To satisfy certain individuals craving for actual historical research and assuming that the webpage here is credible I can now confirm that the assumption that it comes from English turns out to be valid.
Fisher wrote the normal density with $m$ for the mean (see section 12 of his Statistical Methods for Research Workers) until the mid-1930s when he replaced $m$ with $\mu$. The new symbol appears in The Fiducial Argument in Statistical Inference (1935) and it went into the 1936 (sixth) edition of the Statistical Methods for Research Workers.
There is a general rule to use Greek letters for parameters and Latin letters for statistics.
Why $\mu$? Well, the word 'mean' in English starts with M and $\mu$ sounds like M. But also, per Google translate:
- Latin: Media
- French: Moyenne
- Spanish: Media
- German: Mittel
- Dutch: Midden
The use of Greek letters in modern mathematics is basically a consequence of the humanist education. The normal distribution was introduced by Gauß in his celestial mechanics paper "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" in 1809. The parameters $\mu$ and $\sigma$ of the normal distribution are called "Mittelwert" and "Standardabweichung" in German and their Greek letters have the phonetic equivalents "M" and "S" in Latin script.
That may well have been the motivation for giving them those letters. The equivalent Latin (rather than German) terms would likely also start with "M" and "S".