# What is the expected value of $X_i/\|X\|^2$ when $X \sim \mathcal{N}(\mu, \sigma^2I)$

Let $$X$$ be an N-dimensional normal random vector with non-zero mean $$\mu$$ and diagonal covariance matrix $$\sigma^2I$$. I would like to understand if it is possible to derive the expected value of the random variable $$\frac{X_i}{\|X\|^2_2}$$; in other words $$$$\mathbb{E}\left[ \frac{X_i}{\sum_{j=1}^N X^2_j} \right]$$$$ with $$i \in \{1, \dots, N\}$$.

I found similar questions on StackExchange but the assumptions are usually that $$\mu$$ is the zero vector, or that the denominator is the norm $$\|X\|_2$$ (for which one talks about projected normal distribution) instead of the squared norm.

Here, I am only interested in the expectation, and not the distribution itself. A similar expression appears in, for example, the James Stein estimator, and to compute the bias one would need to compute the expectation in question.

• Sep 13, 2022 at 11:46
• Note that conditional on $X_i$, the expectation does not have a closed form expression. Sep 13, 2022 at 13:07
• @User1865345: it seems that the projected Normal considers $X/||X||$ rather than $X/||X||^2$ Sep 13, 2022 at 13:08
• Ah! Yes, @Xi'an. My bad. Sep 13, 2022 at 13:17
• Did you ever find an answer? Oct 10, 2023 at 21:48

In this question I posted, someone gave an answer to your question. 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.