Shift brackets up for fraction in superscript

The code

\documentclass{article}
\begin{document}
$E \left[ e^{\frac{\sigma^2 \lambda^2}{2}} \right]$
\end{document}

produces brackets that symmetric(-ish) around the x-height, so they include the exponent properly, but they look way too low on the bottom. Is there a way to correct this?

• Welcome to TeX.SX! Related/duplicate: tex.stackexchange.com/q/399482/134574 – Phelype Oleinik Nov 14 at 3:21
• It may be related but it's not a duplicate. The other question is about containing a vertically-unbalanced fraction within parentheses. Mine is about a superscript. They might have the same solution, but they aren't the same problem. – japreiss Nov 14 at 3:23
• That's why I didn't vote to close your question as duplicate, but just pointed you to something that might be useful :-) This (and countless many others) was closed as duplicate of this. The questions are completely different but the solution is the same, so I see no problem. – Phelype Oleinik Nov 14 at 3:27
• Doesn't \fixit{[}{e^{\frac{\sigma^2 \lambda^2}{2}}}{]} with \fixit from @PhelypeOleinik's answer solve your problem? – Schrödinger's cat Nov 14 at 5:33

To make the math expression look better, I would not bother with either shifting up the large square brackets or with pulling down the exponent term. Instead, I would use inline-fraction notation to reduce the needlessly large vertical size of the exponent term; this adjustment will also let you use less-prominent square brackets. And, it will make the exponent term easier to read, since TeX will be in script-style instead of in scriptscript-style when it processes the letters \sigma and \lambda in the exponent.

By the way, assuming that "E" denotes the (unconditional) expectations operator, I would typeset the letter in the upright font shape.

Incidentally, I don't get your apparent aversion to \exp notation. It was created for sound typographic reasons.

\documentclass{article}
\usepackage{amsmath} % for "\DeclareMathOperator" macro
\DeclareMathOperator{\E}{E} % expectations operator
\begin{document}
$E \left[ e^{\frac{\sigma^2 \lambda^2}{2}} \right] \quad\text{vs.}\quad \E \bigl[ e^{\sigma^2 \lambda^2/2} \bigr] \quad\text{vs.}\quad \E [ \exp(\sigma^2 \lambda^2/2) ]$
\end{document}
• Thanks, the inline fraction looks decent. I use \mathbb{E} (via a macro) in papers but I was being lazy for this question. I am fine with \exp, I just get annoyed when stack exchange people assume every single question is an "XY problem". – japreiss Nov 14 at 7:32