Formulas or approximations for $\mathbb{E}\left( \frac{X}{\|X\|} \right)$, $X\sim N(\mu, Id)$?

Question

This is a cross-posting of this math SE question.

I want to compute or approximate the following expected value with some analytic expression:

$\mathbb{E}\left( \frac{X}{||X||} \right)$ , where $X \in \mathbb{R}^n$ is a multivariate non-centered gaussian $X\sim N(\mu, \Sigma)$.

For a general $X\sim N(\mu, \Sigma)$ Gaussian, the variable $\frac{X}{||X||}$ is called the general projected normal distribution. This is a complicated distribution, whose moments don't seem to have closed form formulas.

However, I'm wondering whether there is either a useful approximation, or some formula for the case with diagonal/identity covariance. This question comes close to what I need, asking for the expectation $\mathbb{E} \left( \frac{1}{1+\|X\|^2} \right)$ and someone posted an approximation.

One solution I thought of is using a Taylor approximation, in which: $\mathbb{E}\left( \frac{A}{B} \right) \approx \frac{\mathbb{E}(A)}{\mathbb{E}(B)} - \frac{Cov(A,B)}{\mathbb{E}(B)^2} + \frac{\mathbb{E}(A)var(B)}{\mathbb{E}(B)^3}$, where I would have $A = X$ and $B = ||X||$ in the description above. However, I'm not sure there's an expression for $Cov(X, ||X||)$, or whether there's an analytic expression for the variance of the non-centered Chi distribution (i.e. $var(||X||)$).

So, I'm wondering, does the approach above seem valid? Is there expressions for $Cov(X, ||X||)$ and $var(||X||)$ where $X\sim N(\mu, \Sigma)$? Is there some expression or simplification if we assume that $Sigma$ is diagonal or the identity?

Answer 1

Someone provided an exact formula for the expected value in the case of covariance = $\sigma^2 I$ in the cross-post of this question, generalized for different powers of the norm.

The formula is, for $X \sim \mathcal N(\mu,\sigma^2 I)$:

$$ \mathbb E\left( \frac{X}{||X||^k}\right) = \frac{\Gamma\left(\frac{n}{2}+1-\frac{k}{2}\right)}{(2\sigma^2)^{k/2}\Gamma\left(\frac{n}{2}+1\right)}{}_1F_1\left(\frac{k}{2};\frac{n+2}{2};-\frac{|\mu|^2}{2\sigma^2}\right)\,\mu,\tag{$\ast$} $$ where ${}_1F_1(a;b;z)$ is Kummer's confluent hypergeometric function.