Probabilists use the word “expectation” to describe the average of a random variable (which is by the way rather badly defined by the OED:
8. The degree of probability of the occurrence of any contingent event.
1832 WEBSTER s.v., If the chances of receiving or not receiving a hundred dollars..are equal; then..the expectation is worth half the money. 1838 DE MORGAN Ess. Probab. v. (1841) 97 The balance is the average required, and is known by the name of the mathematical expectation. 1848 WHARTON Law Lex., Expectation, in the doctrine of chances, is applied to any contingent event, upon the happening of which some benefit is expected. Ibid., The value of the expectation is..£5.
but let us leave that aside), and the notation is typically
or maybe with a blackboard “E”, or something similar. The notation is also more and more commonly used in some parts of analytic number theory to simplify the writing of (typically) finite averages, with things like
quite common, e.g., in the papers of Green and Tao or the book of additive combinatorics of Tao and Vu.
This is indeed quite convenient, but I find that the LaTeX formatting is often rather ugly, especially in displayed formulas and when there are also integration and summation signs on the line. What I would really like to have is a way to typeset an expectation operator “E” rather like a summation sign “Σ”, with respect to size, and with upper and lower subscripts to indicate the conditions involved. I’ve looked in various collections of LaTeX fonts and symbols without success, and my vanishingly small LaTeX/Metafonting skills are quite insufficient to create such a symbol myself. Does anyone know of a satisfactory solution?