After looking for a built-in expectation symbol in LaTeX, and coming up with none, I’ve defined one. Just add:
% Expectation symbol
DeclareMathOperator*{E}{mathbb{E}}
to your LaTeX preamble, and you’re done. You’ll also need to add usepackage{amsmath}, or in LyX, tick “Use AMS math package” under Document->Settings->Math Options.
Using the starred version of DeclareMathOperator makes sure subscripts go beneath the symbol in display mode.

Thanks for the tip. You need to import
\usepackage{amssymb}in order for this to work.