what is the expected magnitude projection of random unit vector?

Question

For a random unit vector x, write it as $\sum_i a_ie_i$ where $\sum_i a_i^2=1$ and $e_i$ is the basis unit vector. What would be the $E[|a_i|]$?

How to calculate the magnitude $a_i$ relation between different dims?

Answer 1

We can do this integration explicitly using hyperspherical coordinates. Assume $n$ dimensions, and let the angles $\theta_i$, $1 \le i \le n-1$ be the spherical angles, and, without loss of generality, choose $\theta_1$ to be the angle between the random vector and the projected vector. Additionally, let $S_n$ be the $n-1$ dimensional volume of the unit $n$-sphere. Then the resulting integral is $$ E[|a_i|] = \frac{1}{S_n}\int_0^\pi...\int_0^\pi \int_0^{2\pi} |\cos\theta_1|\left(\sin^{n-2}\theta_1 ...\sin\theta_{n-2}\right)d\theta_1...d\theta_{n-2}d\theta_{n-1} \\= \frac{2S_{n-1}}{S_n}\int_0^{\pi/2} \cos\theta_1\sin^{n-2}\theta_1d\theta_1 = \frac{2S_{n-1}}{(n-1)S_n} $$

We also have $S_n = 2\pi^{n/2}/\Gamma(n/2)$, so the expected value is $$ E[|a_i|] = \frac{2 \Gamma(n/2)}{(n-1)\sqrt{\pi}\Gamma(n/2-1/2)} = \frac{2^{n-1}\Gamma(n/2)^2}{\pi \Gamma(n)} $$ For large $n$, this scales as $\sqrt{2/\pi} n^{-1/2}$, which matches the intuitive expectation that $E[|a_i|]\approx n^{-1/2}$.