$\DeclareMathOperator\E{\mathbb E} \DeclareMathOperator\Var{\mathrm{Var}} \newcommand\R{\mathbb R} \DeclareMathOperator\N{\mathcal N} \DeclareMathOperator\tr{\mathrm{tr}}$Suppose $X \sim \N(\mu, \Sigma)$, where $\mathrm{supp}(X) = \R^n$. We can assume $\Sigma$ to be nonsingular. What is $\E \lVert X \rVert$?

If $\Sigma = \sigma^2 I$, then $\lVert X \rVert / \sigma$ follows a noncentral chi distribution, which has known mean (in terms of a generalized Laguerre polynomial). That gives in this case $$ \E \lVert X \rVert = \sigma \sqrt{\frac{\pi}{2}} \; L_{{1/2}}^{{(n/2-1)}}\left({\frac {-\lVert \mu \rVert^{2}}{2 \sigma}}\right)\, ;$$ if we further assume $\mu = 0$ then it becomes a chi distribution and $$ \E \lVert X \rVert = \sigma \sqrt {2} \, \frac{\Gamma((n+1)/2)}{\Gamma(n/2)} .$$

In the general case, we have an easy upper bound via Jensen's inequality. Letting $C^T C = \Sigma$ and $Z \sim \N(0, I)$: \begin{align} \left( \E \lVert X \rVert \right)^2 &< \E \lVert X \rVert^2 \\&= \E \lVert \mu + C Z \rVert^2 \\&= \lVert \mu \rVert^2 + 2 \mu^T C \E Z + \E \tr( Z^T C^T C Z ) \\&= \lVert \mu \rVert^2 + \tr( C^T C \,\E Z Z^T ) \\&= \lVert \mu \rVert^2 + \tr( \Sigma ) ,\end{align} where the inequality is strict since we've assumed $\lVert X \rVert$ is not degenerate.

We also have a lower bound in the same way: since $\lVert \cdot \rVert$ is convex, $\E \lVert X \rVert \ge \lVert \E X \rVert = \lVert \mu \rVert$. (I previously had something more complicated here, based on bounding the variance of the function $z \mapsto \lVert \mu + C z \rVert$, but it gave a worse bound.)

Can we tighten these bounds or, preferably, find an exact expression for $\E \lVert X \rVert$ in the general case? If not, what about the diagonal but non-spherical case, or $\mu = 0$, or other interesting subcases?

There's been some discussion on this site of the distribution of $Y = \lVert X \rVert^2$, particularly here and here. If we have the full distribution of $Y = \lVert X \rVert^2$, we may be able to find $\E \sqrt{Y}$. That distribution is gross, though, and analytical answers seem difficult.


I will not write out a full answer, that will take to much time now. The complete answer to the question can be found in Mathai & Provost: "Quadratic forms in Random Variables", see for example my answer to this question on the Mathematics Stack Exchange site.

The answer is long, complicated and involves special functions such as "the Lauricella function". I don't know if there is an R package for that (I couldn't find one). The answer can be found in that book in the section 3.2b.5: Inverse and Fractional moments in the Nonsingular Normal Case, between pages 56--61.

The distribution of quadratic form in normal variables can be computed with the R package CompQuadForm, maybe you could try numerical integration with the help of that package.

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    $\begingroup$ Cool, that link is a nice trick to turn it into the (square root of) a linear combination of independent chi-squared variables. I don't actually have a need for this, was just made curious by it coming up in this problem, and so numerical integration isn't particularly appealing. I'll check out the book, and am still curious if either the upper or the lower bound can be tightened in a not-overly-complicated way.... $\endgroup$ – djs Apr 5 '15 at 22:28

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